%------------------------------------------------------------------------------
% File     : ITP277_1 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Uniqueness 00371_024092
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0075_VEBT_Uniqueness_00371_024092 [Des22]

% Status   : Theorem
% Rating   : 0.89 v8.2.0, 1.00 v8.1.0
% Syntax   : Number of formulae    : 13755 (2777 unt;2936 typ;   0 def)
%            Number of atoms       : 30370 (9259 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 22899 (3348   ~; 576   |;2472   &)
%                                         (3101 <=>;13402  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   37 (   2 avg)
%            Number of types       :  625 ( 624 usr)
%            Number of type conns  : 2756 (1791   >; 965   *;   0   +;   0  <<)
%            Number of predicates  :  123 ( 120 usr;   3 prp; 0-4 aty)
%            Number of functors    : 2192 (2192 usr; 520 con; 0-8 aty)
%            Number of variables   : 27778 (26819   !; 959   ?;27778   :)
% SPC      : TF0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 16:12:24.391
%------------------------------------------------------------------------------
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    fun_Ex7660298398204468809at_nat: $tType ).

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

tff(ty_n_t__List__Olist_It__List__Olist_It__HOL__Obool_J_J,type,
    list_list_bool: $tType ).

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

tff(ty_n_t__Set__Oset_It__List__Olist_It__HOL__Obool_J_J,type,
    set_list_bool: $tType ).

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

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

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

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

tff(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
    set_list_int: $tType ).

tff(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    list_set_nat: $tType ).

tff(ty_n_t__fun_It__HOL__Obool_Mt__Complex__Ocomplex_J,type,
    fun_bool_complex: $tType ).

tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__HOL__Obool_J,type,
    fun_complex_bool: $tType ).

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

tff(ty_n_t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
    fun_nat_complex: $tType ).

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

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

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

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

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

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

tff(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    set_set_int: $tType ).

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

tff(ty_n_t__List__Olist_It__Extended____Nat__Oenat_J,type,
    list_Extended_enat: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
    set_Product_unit: $tType ).

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

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

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

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

tff(ty_n_t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
    fun_real_bool: $tType ).

tff(ty_n_t__fun_It__HOL__Obool_Mt__Real__Oreal_J,type,
    fun_bool_real: $tType ).

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

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

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

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

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

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

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

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

tff(ty_n_t__fun_It__HOL__Obool_Mt__HOL__Obool_J,type,
    fun_bool_bool: $tType ).

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

tff(ty_n_t__fun_It__Rat__Orat_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__HOL__Obool_J,type,
    fun_num_bool: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
    fun_nat_bool: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
    fun_int_bool: $tType ).

tff(ty_n_t__fun_It__HOL__Obool_Mt__Rat__Orat_J,type,
    fun_bool_rat: $tType ).

tff(ty_n_t__fun_It__HOL__Obool_Mt__Num__Onum_J,type,
    fun_bool_num: $tType ).

tff(ty_n_t__fun_It__HOL__Obool_Mt__Nat__Onat_J,type,
    fun_bool_nat: $tType ).

tff(ty_n_t__fun_It__HOL__Obool_Mt__Int__Oint_J,type,
    fun_bool_int: $tType ).

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

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

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

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

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

tff(ty_n_t__fun_It__Num__Onum_Mt__Num__Onum_J,type,
    fun_num_num: $tType ).

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

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

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

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

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

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

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

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

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

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

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

tff(ty_n_t__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__Option__Ooption_It__HOL__Obool_J,type,
    option_bool: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(ty_n_t__Product____Type__Ounit,type,
    product_unit: $tType ).

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

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

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

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

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

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

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

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

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

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

% Explicit typings (2312)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_bool > fun_bool_bool ).

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

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

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

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

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

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_real_bool > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_list_nat_bool > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_set_nat_bool > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_nat_bool > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_int_bool > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_complex_bool > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_Pr7757163693017731481t_bool > fun_fu6897202038601531823t_bool ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_Ex8563670519792538868t_bool > fun_fu739766798173130341t_bool ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_real_bool > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_list_nat_bool > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_set_nat_bool > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_nat_bool > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_int_bool > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_complex_bool > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_Pr7757163693017731481t_bool > fun_fu6897202038601531823t_bool ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: fun_Ex8563670519792538868t_bool > fun_fu739766798173130341t_bool ).

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

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

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: nat > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: set_complex > fun_set_complex_bool ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: set_Pr1261947904930325089at_nat > fun_se8074476271188067385t_bool ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: set_Extended_enat > fun_se2015444840911768724t_bool ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: set_int > fun_set_int_bool ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_Pr248683899324676269t_bool ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: fun_li8598321315282354825t_bool ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_bo1841576699660343469l_bool ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: int > fun_int_fun_int_bool ).

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

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: set_bool > fun_se689384475314470797l_bool ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: set_real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: set_list_nat > fun_se7035039998181355731t_bool ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: set_set_nat > fun_se1925730536474940935t_bool ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: set_int > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: set_nat > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_real_real ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_int_int ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: set_bool > fun_se689384475314470797l_bool ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: set_real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: set_list_nat > fun_se7035039998181355731t_bool ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: set_set_nat > fun_se1925730536474940935t_bool ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: set_int > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: set_nat > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: set_Pr8693737435421807431at_nat > fun_se3842403354977258381t_bool ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: set_Pr4329608150637261639at_nat > fun_se6392513987399461773t_bool ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: set_Pr8693737435421807431at_nat > fun_Pr446604468264085555t_bool ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: set_Pr4329608150637261639at_nat > fun_Pr1607443439640623155t_bool ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_Pr446604468264085555t_bool > fun_fu4313913773172183267t_bool ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_Pr1607443439640623155t_bool > fun_fu224428745893945571t_bool ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: produc859450856879609959at_nat > fun_Pr446604468264085555t_bool ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: produc3843707927480180839at_nat > fun_Pr1607443439640623155t_bool ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: set_bool > fun_bool_bool ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: set_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: set_list_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: set_set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: set_nat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: set_int > fun_int_bool ).

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

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

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

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: fun_int_bool > fun_int_bool ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: set_bool > fun_fu5912831971618039257l_bool ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: set_real > fun_fu4470565429337120259l_bool ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: set_bool > fun_fu6847067296440702169l_bool ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: set_real > fun_fu66453571216240387l_bool ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: set_bool > fun_fu2638809437031423321l_bool ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: set_real > fun_fu6440467031773438595l_bool ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: set_bool > fun_fu3921281471375230497l_bool ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: set_real > fun_fu8829865716563604795l_bool ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: set_nat > fun_fu4970875099359191043t_bool ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: set_nat > fun_fu7165551060251750659t_bool ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_Pr7757163693017731481t_bool > fun_Pr248683899324676269t_bool ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_list_nat_bool > fun_li8598321315282354825t_bool ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_set_nat_bool > fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: fun_real_bool > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: fun_bool_bool > fun_bo1841576699660343469l_bool ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_int_bool > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: fun_Pr7757163693017731481t_bool > fun_Pr248683899324676269t_bool ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_list_nat_bool > fun_li8598321315282354825t_bool ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: fun_set_nat_bool > fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_real_bool > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: fun_bool_bool > fun_bo1841576699660343469l_bool ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: fun_int_bool > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: product_prod_nat_nat > fun_fu6897202038601531823t_bool ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: bool > fun_fu7925023953348749699l_bool ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: real > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: list_nat > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_nat > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: product_prod_nat_nat > fun_se3305628495172992525t_bool ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: bool > fun_se689384475314470797l_bool ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: list_nat > fun_se7035039998181355731t_bool ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: set_nat > fun_se1925730536474940935t_bool ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: nat > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: int > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: fun_real_bool ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_bool_bool ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_int_bool ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_real_bool > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_list_nat_bool > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_set_nat_bool > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_int_bool > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: fun_nat_bool > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: fun_Pr446604468264085555t_bool > fun_fu4313913773172183267t_bool ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_Pr1607443439640623155t_bool > fun_fu224428745893945571t_bool ).

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

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: set_bool > fun_fu7925023953348749699l_bool ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: set_real > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: set_list_nat > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: set_set_nat > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: set_nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: set_int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: set_Pr6588086440996610945on_nat > fun_op7792301606653096457t_bool ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: set_Pr7459493094073627847at_nat > fun_se5010674451832123565t_bool ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: set_Pr4329608150637261639at_nat > fun_se1534075524646452397t_bool ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: set_Pr958786334691620121nt_int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: set_bool > fun_se689384475314470797l_bool ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: set_real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: set_list_nat > fun_se7035039998181355731t_bool ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: set_set_nat > fun_se1925730536474940935t_bool ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: set_int > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: set_nat > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: set_Pr1261947904930325089at_nat > fun_se3305628495172992525t_bool ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: set_Pr1261947904930325089at_nat > fun_Pr7757163693017731481t_bool ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: real > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: rat > fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: set_nat > fun_nat_fun_nat_bool ).

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

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_na6801867027684862029l_bool ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: set_Pr1261947904930325089at_nat > fun_na5265545386895126880t_bool ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: set_Extended_enat > fun_na4695037785009495611t_bool ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: set_list_nat > fun_na5915592536692849161t_bool ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: set_set_nat > fun_na5144080005386289891t_bool ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: set_Pr1261947904930325089at_nat > fun_na5265545386895126880t_bool ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: set_Extended_enat > fun_na4695037785009495611t_bool ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_re4238438383274954274t_real ).

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

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

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_nat_real ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_nat_int ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: fun_nat_bool > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: fun_nat_bool > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_nat_bool > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_nat_real > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_nat_complex > fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_nat_int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_nat_real > fun_nat_real ).

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

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

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

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

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

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: set_bool > fun_fu8231870962471999726l_bool ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: set_bool > fun_fu2959318322722153305l_bool ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: set_real > fun_fu914841811607101702l_bool ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: set_real > fun_fu7100649987335273288l_bool ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: set_nat > fun_fu4346542433904371390t_bool ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: set_nat > fun_fu8382797250367883996t_bool ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: set_int > fun_fu7152481525987436038t_bool ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: set_int > fun_fu5129045444369776328t_bool ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: set_complex > fun_fu5519326811792979714x_bool ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: set_complex > fun_fu3191020621175015362x_bool ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: set_Extended_enat > fun_fu7727529105362479790t_bool ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: set_Extended_enat > fun_fu4596718790507443722t_bool ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: set_bool > fun_fu2238024911088387052l_bool ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: set_bool > fun_fu9084992954297690841l_bool ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: set_bool > fun_fu2238024911088387052l_bool ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: set_bool > fun_fu9084992954297690841l_bool ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: set_complex > fun_fu5995210064384465734x_bool ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: set_Extended_enat > fun_fu3738011059562840620t_bool ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: set_Extended_enat > fun_fu5527893311006036234t_bool ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: set_bool > fun_fu6678276313180071960l_bool ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: set_bool > fun_fu8303122171908583641l_bool ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: set_real > fun_fu2277326744902863452l_bool ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: set_real > fun_fu6750725090257165812l_bool ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: set_nat > fun_fu3052598110916186900t_bool ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: set_nat > fun_fu2971308621445215880t_bool ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: set_int > fun_fu337402001079966300t_bool ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: set_int > fun_fu163127709195666036t_bool ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: bool > fun_fu4678014405698904899l_real ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: extended_enat > fun_fu7336479014630192387t_real ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: bool > fun_fu7257246399608679363ol_rat ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: bool > fun_fu4678014405698904899l_real ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: extended_enat > fun_fu7336479014630192387t_real ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: bool > fun_fu7257246399608679363ol_rat ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: complex > fun_fu2990765671291504771ex_rat ).

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

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: set_nat > fun_fu2988988732329857685t_bool ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: set_nat > fun_fu8646012520862679939ol_nat ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_bool_fun_nat_nat > fun_nat_fun_bool_nat ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: set_bool > fun_fu1759362394388581888at_nat ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: set_nat > fun_fu9073671928554219267t_bool ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: set_nat > fun_fu6463170612142064108al_nat ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_real_fun_nat_nat > fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: set_real > fun_fu3370445115705697296at_nat ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: set_nat > fun_fu2583778531842586371t_bool ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: set_nat > fun_fu8221675813083619692nt_nat ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: set_int > fun_fu7719728765329718531t_bool ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: set_int > fun_fu5002972218919249680at_nat ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: set_nat > fun_fu5088574307552523139t_bool ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: set_nat > fun_fu4637029388598527722ex_nat ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_co636942507509095880at_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: set_complex > fun_fu1598310740840436355x_bool ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: set_complex > fun_fu1817986637694864396at_nat ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: set_nat > fun_fu8451253408283314807t_bool ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: set_nat > fun_fu8372565977779661586at_nat ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_Ex7123704652682714774at_nat > fun_na40478436225343698at_nat ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: set_Extended_enat > fun_fu2429753552268211515t_bool ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: set_Extended_enat > fun_fu2544567023866575296at_nat ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: set_complex > fun_fu2900838443003976153x_bool ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: set_complex > fun_fu882372458115418499omplex ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_bo6990880009230973955omplex > fun_co2083647329047648387omplex ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: set_bool > fun_fu3808756750993963034omplex ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: set_complex > fun_fu4189668387880940419x_bool ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: set_complex > fun_fu3241646296630235056omplex ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_re7208826350674946562omplex > fun_co286204877037288194omplex ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: set_real > fun_fu7272995576155203506omplex ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: set_complex > fun_fu8281188428872102787x_bool ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: set_complex > fun_fu5276431085538847372omplex ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_na1591161823130732966omplex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: set_nat > fun_fu2548079958729413763t_bool ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: set_nat > fun_fu5286008751397070186omplex ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: set_complex > fun_fu3391973010043818627x_bool ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: set_complex > fun_fu2147946303232500784omplex ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: fun_in1307110748760623106omplex > fun_co7673900812888889602omplex ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: set_int > fun_fu8949065344085950851t_bool ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: set_int > fun_fu2298032621795249330omplex ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: set_complex > fun_fu1223225621849392571x_bool ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: set_complex > fun_fu8345204294023322898omplex ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_Ex6753287174829011282omplex > fun_co6680409715755339346omplex ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: set_Extended_enat > fun_fu8436912864957114555t_bool ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: set_Extended_enat > fun_fu6154561449118725082omplex ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_complex_complex > fun_complex_real ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_bool_real > fun_fu8399412978111079276l_real ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: set_complex > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_complex_real > fun_fu1911030855436031788x_real ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: set_Extended_enat > fun_fu739766798173130341t_bool ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_Ex1451620480498224165t_real > fun_fu4696490002264930058t_real ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: fun_bool_rat > fun_fu7923017422675304344ol_rat ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_real_rat > fun_fu7249360421009165912al_rat ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_nat_rat > fun_fu2739651132436891864at_rat ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_int_rat > fun_fu2671659548940339672nt_rat ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_complex_rat > fun_fu4947971316162376536ex_rat ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_nat_real > fun_se2100927751953696370t_real ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_real_real > fun_se5601468920065002932l_real ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_bool_real > fun_se1633681991067242466l_real ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_int_real > fun_se7886327339988957492t_real ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_complex_real > fun_se7589301288977092664x_real ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_Ex1451620480498224165t_real > fun_se6362295375323711138t_real ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_real_rat > fun_se2428050584324877792al_rat ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_bool_rat > fun_se7741389517691119374ol_rat ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_int_rat > fun_se937733209267607904nt_rat ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_complex_rat > fun_se8588693959369716324ex_rat ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_Ex4152069839811072081at_rat > fun_se1920680211210075598at_rat ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_real_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: fun_int_real > fun_int_bool ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: fun_complex_real > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: fun_Ex1451620480498224165t_real > fun_Ex8563670519792538868t_bool ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_real_rat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_int_rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_complex_rat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_Ex4152069839811072081at_rat > fun_Ex8563670519792538868t_bool ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_real_nat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: fun_int_nat > fun_int_bool ).

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

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_real_bool > fun_fu9132278742410140537l_real ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: fun_int_bool > fun_fu4328765270694171897t_real ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_complex_bool > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: fun_Ex8563670519792538868t_bool > fun_fu3686806697881729723t_real ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_real_bool > fun_fu7187499828406537085al_rat ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: fun_int_bool > fun_fu1617438229336688381nt_rat ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: fun_complex_bool > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: fun_Ex8563670519792538868t_bool > fun_fu7576661262455177319at_rat ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_real_bool > fun_fu3768230810189712741al_nat ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: fun_int_bool > fun_fu973771059875708133nt_nat ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: int > fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: complex > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: extended_enat > fun_fu3686806697881729723t_real ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: complex > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: extended_enat > fun_fu7576661262455177319at_rat ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: complex > fun_fu2513798379308225899ex_nat ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: extended_enat > fun_fu8366861321737653343at_nat ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: complex > fun_fu4692242605864086655ex_int ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: extended_enat > fun_fu2469135700067826619at_int ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: real > fun_fu9132278742410140537l_real ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: real > fun_fu7187499828406537085al_rat ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: nat > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: nat > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_bo7282069421189217543nteger > fun_fu624093908337759363nteger ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: fun_bool_real > fun_fu4678014405698904899l_real ).

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

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_real_rat > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_bool_rat > fun_fu7257246399608679363ol_rat ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: fun_nat_nat ).

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

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

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

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: extended_enat > fun_Ex2441982661509597004d_enat ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: int > fun_int_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: real > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: set_bool > fun_fu8539566175054216963l_bool ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: set_bool > fun_fu8539566175054216963l_bool ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: set_bool > fun_fu7353774216694237776ol_nat ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: set_bool > fun_fu8853735418707552473l_bool ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: set_real > fun_fu6884438414433897309l_bool ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: set_real > fun_fu4541298345424915491ol_nat ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: set_real > fun_fu5795206914620844761l_bool ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: set_bool > fun_fu7764503918894499925l_bool ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: set_bool > fun_fu7286723868770535586al_nat ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: set_real > fun_fu6442698403317780995l_bool ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: set_real > fun_fu6442698403317780995l_bool ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: set_real > fun_fu8273873231336023293al_nat ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: set_int > fun_fu3214323050464676829t_bool ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: set_int > fun_fu8444598815406890787ol_nat ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: set_int > fun_fu3139726878529933315t_bool ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: set_int > fun_fu5144190403709078909al_nat ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: set_complex > fun_fu7148735174882109861ol_nat ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: set_complex > fun_fu7828569628974167419al_nat ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: set_Extended_enat > fun_fu3218030382291066435t_bool ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: set_Extended_enat > fun_fu702193231572143775ol_nat ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: set_Extended_enat > fun_fu6580014649061146295t_bool ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: set_Extended_enat > fun_fu5726332634718240019al_nat ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_bool_nat > fun_fu560003646518074819ol_nat ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: set_nat > fun_fu9217928336980114631ol_nat ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: set_nat > fun_fu6241273384353077465al_nat ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_int_int ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_nat_complex > fun_co3258106519541737638omplex ).

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

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

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: fun_num_fun_num_bool ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: real > fun_na5225841814499866095t_real ).

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

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_int_int ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: fun_real_real ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: fun_nat_fun_nat_bool ).

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

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_real_bool ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: fun_re8661203372455305201t_bool ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: nat > fun_na5042103099923536755t_bool ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_real_real ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: fun_nat_real > fun_fu6899955116716589935t_real ).

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

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_nat_real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: real > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: nat > fun_li8232715659020003717t_bool ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: set_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: set_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: real > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_re5512370375968561900l_real ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: real > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: fun_re3844104906058862192omplex ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: code_integer > fun_Co3025358997185726883r_bool ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: fun_nat_real > fun_fu3106432010075488275t_real ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: fun_real_bool ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: fun_co2286642472385812463l_bool ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: rat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: rat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: rat > fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: fun_fun_nat_rat_bool ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: fun_Ex7682446853002821117t_bool ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: fun_Ex7682446853002821117t_bool ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: real > fun_re8658731967611918585t_real ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: set_Extended_enat > fun_Ex8563670519792538868t_bool ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: fun_na819858167508000762st_nat ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: fun_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: fun_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: fun_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: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__HOL__Obool_001t__HOL__Obool,type,
    bNF_re9007465950226028919l_bool: ( fun_int_fun_int_bool * fun_bo1841576699660343469l_bool ) > fun_fu7925731147167280621l_bool ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__HOL__Obool_J_001t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
    bNF_re5780769757191362231t_bool: ( fun_int_fun_int_bool * fun_fu7925731147167280621l_bool * fun_int_fun_int_bool * fun_int_fun_int_bool ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__HOL__Obool_001t__HOL__Obool,type,
    bNF_re6223507645442431167l_bool: ( fun_nat_fun_nat_bool * fun_bo1841576699660343469l_bool ) > fun_fu5827305053654988013l_bool ).

tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
    bNF_re1480548845665215743t_bool: ( fun_nat_fun_nat_bool * fun_fu5827305053654988013l_bool * fun_nat_fun_nat_bool * fun_nat_fun_nat_bool ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__HOL__Obool_001t__HOL__Obool,type,
    bNF_re1394333895916556125l_bool: ( fun_Pr3099820731930151597t_bool * fun_bo1841576699660343469l_bool * fun_Pr803113274115125473t_bool * fun_Pr803113274115125473t_bool ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__HOL__Obool_001t__HOL__Obool,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__HOL__Obool_001t__HOL__Obool,type,
    bNF_re1498568104593929552l_bool: ( fun_Pr5932607294251420486t_bool * fun_bo1841576699660343469l_bool ) > fun_fu813382667476268716l_bool ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__HOL__Obool_J_001t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
    bNF_re2632563231991741785t_bool: ( fun_Pr5932607294251420486t_bool * fun_fu813382667476268716l_bool * fun_Pr248683899324676269t_bool * fun_int_fun_int_bool ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__HOL__Obool_001t__HOL__Obool,type,
    bNF_re3189209160384574813l_bool: ( fun_Pr248683899324676269t_bool * fun_bo1841576699660343469l_bool ) > fun_fu8778301040783821591l_bool ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__HOL__Obool_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__HOL__Obool_J,type,
    bNF_re2219331435504962769t_bool: ( fun_Pr248683899324676269t_bool * fun_fu8778301040783821591l_bool * fun_Pr248683899324676269t_bool * fun_Pr248683899324676269t_bool ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001t__HOL__Obool_001t__HOL__Obool,type,
    bNF_re1458715823057739248l_bool: ( fun_fu990398353789569848l_bool * fun_bo1841576699660343469l_bool * fun_fun_nat_rat_bool * fun_real_bool ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__HOL__Obool_001t__HOL__Obool,type,
    bNF_re2717234250851183977l_bool: ( fun_fu2636726645394866687t_bool * fun_bo1841576699660343469l_bool * fun_fun_nat_rat_bool * fun_fun_nat_rat_bool ) > $o ).

tff(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
    bNF_We3818239936649020644el_nat: fun_se8074476271188067385t_bool ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: fun_nat_fun_nat_nat ).

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

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

tff(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
    gbinomial_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
    gbinomial_real: fun_re4238438383274954274t_real ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr3099820731930151597t_bool ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: nat > fun_int_fun_int_int ).

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__Int__Oint,type,
    bit_se725231765392027082nd_int: fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
    bit_se8570568707652914677it_nat: fun_nat_fun_nat_nat ).

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

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__Int__Oint,type,
    bit_se1409905431419307370or_int: fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
    bit_se1412395901928357646or_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
    bit_se547839408752420682it_nat: fun_nat_fun_nat_nat ).

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

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
    bit_se7882103937844011126it_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
    bit_se2925701944663578781it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
    bit_se4203085406695923979it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
    bit_se4205575877204974255it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
    bit_se6526347334894502574or_int: fun_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__Int__Oint,type,
    bit_se1146084159140164899it_int: fun_int_fun_nat_bool ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
    bit_se1148574629649215175it_nat: fun_nat_fun_nat_bool ).

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

tff(sy_c_COMBB_001t__HOL__Obool_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    cOMBB_3745307990312268622T_VEBT: ( fun_bool_bool * fun_VEBT_VEBT_bool ) > fun_VEBT_VEBT_bool ).

tff(sy_c_COMBB_001t__HOL__Obool_001t__fun_It__HOL__Obool_Mt__HOL__Obool_J_001t__VEBT____Definitions__OVEBT,type,
    cOMBB_3438363824644082366T_VEBT: ( fun_bo1841576699660343469l_bool * fun_VEBT_VEBT_bool ) > fun_VE2066785035971159626l_bool ).

tff(sy_c_COMBB_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    cOMBB_2885587777813823575T_VEBT: ( fun_fu7168253649841883962l_bool * fun_VE3763956218935105199t_bool ) > fun_VEBT_VEBT_bool ).

tff(sy_c_COMBC_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_001t__HOL__Obool,type,
    cOMBC_771164880915498576T_bool: ( fun_VE7346385682144900493T_bool * set_VEBT_VEBT ) > fun_VEBT_VEBT_bool ).

tff(sy_c_COMBS_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool_001t__HOL__Obool,type,
    cOMBS_6222329278419335549l_bool: ( fun_VE2066785035971159626l_bool * fun_VEBT_VEBT_bool ) > fun_VEBT_VEBT_bool ).

tff(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > produc7417142470700560111r_bool ).

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

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

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

tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: fun_int_Code_integer ).

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

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

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Extended____Nat__Oenat,type,
    comple2295165028678016749d_enat: set_Extended_enat > extended_enat ).

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

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

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
    complete_Inf_Inf_nat: fun_set_nat_nat ).

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

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Extended____Nat__Oenat,type,
    comple4398354569131411667d_enat: set_Extended_enat > extended_enat ).

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    lattic6202137068483785622ol_rat: fun_bool_rat > fun_set_bool_bool ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__HOL__Obool_001t__Real__Oreal,type,
    lattic5237676299781464426l_real: fun_bool_real > fun_set_bool_bool ).

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

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Real__Oreal,type,
    lattic2675449441010098035t_real: ( fun_int_real * set_int ) > int ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Rat__Orat,type,
    lattic6811802900495863747at_rat: fun_nat_rat > fun_set_nat_nat ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Real__Oreal,type,
    lattic488527866317076247t_real: fun_nat_real > fun_set_nat_nat ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Num__Onum,type,
    lattic1613168225601753569al_num: ( fun_real_num * set_real ) > real ).

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

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Real__Oreal,type,
    lattic8440615504127631091l_real: ( fun_real_real * set_real ) > real ).

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

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

tff(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
    at_infinity_real: filter_real ).

tff(sy_c_List_Oappend_001t__Int__Oint,type,
    append_int: ( list_int * list_int ) > list_int ).

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

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

tff(sy_c_List_Oconcat_001t__Nat__Onat,type,
    concat_nat: list_list_nat > list_nat ).

tff(sy_c_List_Oconcat_001t__VEBT____Definitions__OVEBT,type,
    concat_VEBT_VEBT: list_list_VEBT_VEBT > list_VEBT_VEBT ).

tff(sy_c_List_Oenumerate_001t__HOL__Obool,type,
    enumerate_bool: ( nat * list_bool ) > list_P6781292141273955916t_bool ).

tff(sy_c_List_Oenumerate_001t__Int__Oint,type,
    enumerate_int: ( nat * list_int ) > list_P3521021558325789923at_int ).

tff(sy_c_List_Oenumerate_001t__Nat__Onat,type,
    enumerate_nat: ( nat * list_nat ) > list_P6011104703257516679at_nat ).

tff(sy_c_List_Oenumerate_001t__VEBT____Definitions__OVEBT,type,
    enumerate_VEBT_VEBT: ( nat * list_VEBT_VEBT ) > list_P5647936690300460905T_VEBT ).

tff(sy_c_List_Ofind_001t__HOL__Obool,type,
    find_bool: ( fun_bool_bool * list_bool ) > option_bool ).

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

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

tff(sy_c_List_Ofind_001t__Num__Onum,type,
    find_num: ( fun_num_bool * list_num ) > option_num ).

tff(sy_c_List_Ofind_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    find_P8199882355184865565at_nat: ( fun_Pr7757163693017731481t_bool * list_P6011104703257516679at_nat ) > option4927543243414619207at_nat ).

tff(sy_c_List_Ofind_001t__Real__Oreal,type,
    find_real: ( fun_real_bool * list_real ) > option_real ).

tff(sy_c_List_Ofind_001t__Set__Oset_It__Nat__Onat_J,type,
    find_set_nat: ( fun_set_nat_bool * list_set_nat ) > option_set_nat ).

tff(sy_c_List_Ofind_001t__VEBT____Definitions__OVEBT,type,
    find_VEBT_VEBT: ( fun_VEBT_VEBT_bool * list_VEBT_VEBT ) > option_VEBT_VEBT ).

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

tff(sy_c_List_Olast_001t__Nat__Onat,type,
    last_nat: list_nat > nat ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
    linord2614967742042102400et_nat: set_nat > list_nat ).

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

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

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

tff(sy_c_List_Olist_OCons_001t__Num__Onum,type,
    cons_num: ( num * list_num ) > list_num ).

tff(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    cons_P6512896166579812791at_nat: ( product_prod_nat_nat * list_P6011104703257516679at_nat ) > list_P6011104703257516679at_nat ).

tff(sy_c_List_Olist_OCons_001t__Real__Oreal,type,
    cons_real: ( real * list_real ) > list_real ).

tff(sy_c_List_Olist_OCons_001t__Set__Oset_It__Nat__Onat_J,type,
    cons_set_nat: ( set_nat * list_set_nat ) > list_set_nat ).

tff(sy_c_List_Olist_OCons_001t__VEBT____Definitions__OVEBT,type,
    cons_VEBT_VEBT: ( vEBT_VEBT * list_VEBT_VEBT ) > list_VEBT_VEBT ).

tff(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

tff(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

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

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

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

tff(sy_c_List_Olist_Oset_001t__Extended____Nat__Oenat,type,
    set_Extended_enat2: list_Extended_enat > set_Extended_enat ).

tff(sy_c_List_Olist_Oset_001t__HOL__Obool,type,
    set_bool2: list_bool > set_bool ).

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

tff(sy_c_List_Olist_Oset_001t__List__Olist_It__HOL__Obool_J,type,
    set_list_bool2: list_list_bool > set_list_bool ).

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

tff(sy_c_List_Olist_Oset_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    set_list_VEBT_VEBT2: list_list_VEBT_VEBT > set_list_VEBT_VEBT ).

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

tff(sy_c_List_Olist_Oset_001t__Num__Onum,type,
    set_num2: list_num > set_num ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__HOL__Obool_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr3250499561091541003T_VEBT: ( list_P5912569763298001674T_VEBT * nat ) > produc8408751633324799108T_VEBT ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__HOL__Obool_J,type,
    nth_Pr6495399312670224281t_bool: ( list_P6321009189627309296t_bool * nat ) > produc4876081531382960352t_bool ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    nth_Pr4439495888332055232nt_int: ( list_P5707943133018811711nt_int * nat ) > product_prod_int_int ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
    nth_Pr8617346907841251940nt_nat: ( list_P8198026277950538467nt_nat * nat ) > product_prod_int_nat ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr3474266648193625910T_VEBT: ( list_P7524865323317820941T_VEBT * nat ) > produc1531783533982839933T_VEBT ).

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

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    nth_Pr3440142176431000676at_int: ( list_P3521021558325789923at_int * nat ) > product_prod_nat_int ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    nth_Pr7617993195940197384at_nat: ( list_P6011104703257516679at_nat * nat ) > product_prod_nat_nat ).

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

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

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

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

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr4953567300277697838T_VEBT: ( list_P7413028617227757229T_VEBT * nat ) > produc8243902056947475879T_VEBT ).

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

tff(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
    nth_set_nat: ( list_set_nat * nat ) > set_nat ).

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

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

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

tff(sy_c_List_Oproduct_001t__Int__Oint_001t__HOL__Obool,type,
    product_int_bool: ( list_int * list_bool ) > list_P6321009189627309296t_bool ).

tff(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
    product_int_int: ( list_int * list_int ) > list_P5707943133018811711nt_int ).

tff(sy_c_List_Oproduct_001t__Int__Oint_001t__Nat__Onat,type,
    product_int_nat: ( list_int * list_nat ) > list_P8198026277950538467nt_nat ).

tff(sy_c_List_Oproduct_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    produc662631939642741121T_VEBT: ( list_int * list_VEBT_VEBT ) > list_P7524865323317820941T_VEBT ).

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

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

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

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

tff(sy_c_List_OremoveAll_001t__HOL__Obool,type,
    removeAll_bool: ( bool * list_bool ) > list_bool ).

tff(sy_c_List_OremoveAll_001t__Int__Oint,type,
    removeAll_int: ( int * list_int ) > list_int ).

tff(sy_c_List_OremoveAll_001t__Nat__Onat,type,
    removeAll_nat: ( nat * list_nat ) > list_nat ).

tff(sy_c_List_OremoveAll_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    remove3673390508374433037at_nat: ( product_prod_nat_nat * list_P6011104703257516679at_nat ) > list_P6011104703257516679at_nat ).

tff(sy_c_List_OremoveAll_001t__Real__Oreal,type,
    removeAll_real: ( real * list_real ) > list_real ).

tff(sy_c_List_OremoveAll_001t__Set__Oset_It__Nat__Onat_J,type,
    removeAll_set_nat: ( set_nat * list_set_nat ) > list_set_nat ).

tff(sy_c_List_OremoveAll_001t__VEBT____Definitions__OVEBT,type,
    removeAll_VEBT_VEBT: ( vEBT_VEBT * list_VEBT_VEBT ) > list_VEBT_VEBT ).

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

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

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

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

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

tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
    sorted_wrt_int: ( fun_int_fun_int_bool * list_int ) > $o ).

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

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

tff(sy_c_List_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_Pr3099820731930151597t_bool ).

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

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

tff(sy_c_List_Ozip_001t__Int__Oint_001t__HOL__Obool,type,
    zip_int_bool: ( list_int * list_bool ) > list_P6321009189627309296t_bool ).

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

tff(sy_c_List_Ozip_001t__Int__Oint_001t__Nat__Onat,type,
    zip_int_nat: ( list_int * list_nat ) > list_P8198026277950538467nt_nat ).

tff(sy_c_List_Ozip_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    zip_int_VEBT_VEBT: ( list_int * list_VEBT_VEBT ) > list_P7524865323317820941T_VEBT ).

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

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

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

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

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

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

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

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tff(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: fun_rat_fun_rat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__HOL__Obool_Mt__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__HOL__Obool,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__HOL__Obool_Mt__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__HOL__Obool_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
    power_power_int: int > fun_nat_int ).

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

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

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__HOL__Obool,type,
    produc5135875973608410785r_bool: ( code_integer * bool ) > produc7417142470700560111r_bool ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Int__Oint,type,
    produc8871513391732212460ol_int: ( bool * int ) > produc5612625445546783218ol_int ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    produc9074556814250467700T_VEBT: ( bool * vEBT_VEBT ) > produc8408751633324799108T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__HOL__Obool,type,
    produc788913908665201810t_bool: ( int * bool ) > produc4876081531382960352t_bool ).

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

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Nat__Onat,type,
    product_Pair_int_nat: ( int * nat ) > product_prod_int_nat ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    produc3329399203697025711T_VEBT: ( int * vEBT_VEBT ) > produc1531783533982839933T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__HOL__Obool,type,
    produc3913204875133826030t_bool: ( nat * bool ) > produc8000372497851584572t_bool ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
    product_Pair_nat_int: ( nat * int ) > product_prod_nat_int ).

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

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc487386426758144856at_nat: ( nat * product_prod_nat_nat ) > produc7248412053542808358at_nat ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc599794634098209291T_VEBT: ( nat * vEBT_VEBT ) > produc8025551001238799321T_VEBT ).

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

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

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

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

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    produc3790773574474814305omplex: ( set_complex * set_complex ) > produc8064648209034914857omplex ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Extended____Nat__Oenat_J_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    produc6639060556116774935d_enat: ( set_Extended_enat * set_Extended_enat ) > produc1621487020699730983d_enat ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
    produc6363374080413544029et_int: ( set_int * set_int ) > produc2115011035271226405et_int ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc4532415448927165861et_nat: ( set_nat * set_nat ) > produc7819656566062154093et_nat ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc2922128104949294807at_nat: ( set_Pr1261947904930325089at_nat * set_Pr1261947904930325089at_nat ) > produc3843707927480180839at_nat ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    produc9060074326276436823at_nat: ( set_Pr4329608150637261639at_nat * set_Pr4329608150637261639at_nat ) > produc1319942482725812455at_nat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
    produc3035007008238225530T_bool: ( vEBT_VEBT * bool ) > produc6448083994118892042T_bool ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc736041933913180425BT_int: ( vEBT_VEBT * int ) > produc4894624898956917775BT_int ).

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

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc537772716801021591T_VEBT: ( vEBT_VEBT * vEBT_VEBT ) > produc8243902056947475879T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc2004789812759622991on_nat: ( fun_nat_fun_nat_bool * produc4953844613479565601on_nat ) > produc3700678068921409119on_nat ).

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

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

tff(sy_c_Product__Type_OPair_001t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_Mt__HOL__Obool_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    produc8433281988922931063on_num: ( fun_num_fun_num_bool * produc3447558737645232053on_num ) > produc6167219322466932359on_num ).

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

tff(sy_c_Product__Type_OPair_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__HOL__Obool_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc5961576194287291153at_nat: ( fun_Pr248683899324676269t_bool * produc6121120109295599847at_nat ) > produc533602882674032673at_nat ).

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

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

tff(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    produc1553301316500091796er_int: ( fun_Co1092983284983208416er_int * produc8923325533196201883nteger ) > int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6916734918728496179nteger: ( fun_Co2470855966376689427nteger * produc8923325533196201883nteger ) > produc8923325533196201883nteger ).

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__HOL__Obool_J,type,
    produc6117575242473779287r_bool: ( fun_Co3025358997185726883r_bool * produc8923325533196201883nteger ) > produc7417142470700560111r_bool ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4245557441103728435nt_int: ( fun_in1696471567231710611nt_int * product_prod_int_int ) > product_prod_int_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__HOL__Obool,type,
    produc8955392012077580105t_bool: fun_nat_fun_nat_bool > fun_Pr7757163693017731481t_bool ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    produc2761476792215241774st_nat: ( fun_na819858167508000762st_nat * product_prod_nat_nat ) > list_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__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_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__HOL__Obool_J,type,
    produc5023667602222594023t_bool: fun_na4786964484885125043t_bool > fun_Pr248683899324676269t_bool ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001t__HOL__Obool,type,
    produc5257810830037637649l_bool: fun_re1255401033344146673l_bool > fun_Pr9030630356418945121l_bool ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
    product_fst_int_int: product_prod_int_int > int ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
    product_snd_int_int: product_prod_int_int > int ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Product__Type_Ounit_OAbs__unit,type,
    product_Abs_unit: fun_bo1891391082718140537t_unit ).

tff(sy_c_Product__Type_Ounit_ORep__unit,type,
    product_Rep_unit: fun_Pr1572506220757514509t_bool ).

tff(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: product_prod_int_int > rat ).

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

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: fun_ra4503298312962071866nt_int ).

tff(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
    field_5140801741446780682s_real: set_real ).

tff(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
    field_7254667332652039916t_real: fun_rat_real ).

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

tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun_Pr3145601080792579042t_bool ).

tff(sy_c_Rat_Opositive,type,
    positive: fun_rat_bool ).

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

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun_Pr3099820731930151597t_bool ).

tff(sy_c_Real_OReal,type,
    real2: fun_nat_rat > real ).

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

tff(sy_c_Real_Opcr__real,type,
    pcr_real: fun_fu990398353789569848l_bool ).

tff(sy_c_Real_Opositive,type,
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tff(sy_c_Real_Orealrel,type,
    realrel: fun_fu2636726645394866687t_bool ).

tff(sy_c_Real_Orep__real,type,
    rep_real: fun_real_fun_nat_rat ).

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

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
    real_V975177566351809787t_real: fun_re7681423839302404990l_real ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
    real_V1022390504157884413omplex: fun_complex_real ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
    real_V7735802525324610683m_real: fun_real_real ).

tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
    real_V4546457046886955230omplex: fun_real_complex ).

tff(sy_c_Relation_OField_001t__Nat__Onat,type,
    field_nat: set_Pr1261947904930325089at_nat > set_nat ).

tff(sy_c_Relation_OId_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Relation_Otrans_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    trans_4347625901269045472at_nat: set_Pr8693737435421807431at_nat > $o ).

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

tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: fun_int_fun_int_int ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
    divide_divide_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
    divide_divide_rat: fun_rat_fun_rat_rat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
    divide_divide_real: fun_re7681423839302404990l_real ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: fun_int_fun_int_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Rings_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: fun_int_fun_int_int ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
    modulo_modulo_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Ounit__factor__class_Ounit__factor_001t__Nat__Onat,type,
    unit_f2748546683901255202or_nat: fun_nat_nat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: fun_bool_int ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
    zero_n2687167440665602831ol_nat: fun_bool_nat ).

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,
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tff(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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tff(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osummable_001t__Int__Oint,type,
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tff(sy_c_Series_Osummable_001t__Nat__Onat,type,
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tff(sy_c_Series_Osummable_001t__Real__Oreal,type,
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tff(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osums_001t__Int__Oint,type,
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tff(sy_c_Series_Osums_001t__Nat__Onat,type,
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tff(sy_c_Series_Osums_001t__Real__Oreal,type,
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tff(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
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tff(sy_c_Set_OCollect_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Set_OCollect_001t__HOL__Obool,type,
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tff(sy_c_Set_OCollect_001t__Int__Oint,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__HOL__Obool_J,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Set_OCollect_001t__Nat__Onat,type,
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tff(sy_c_Set_OCollect_001t__Num__Onum,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
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tff(sy_c_Set_OCollect_001t__Rat__Orat,type,
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tff(sy_c_Set_OCollect_001t__Real__Oreal,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_OPow_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
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tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
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tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
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tff(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
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tff(sy_c_Set_Oinsert_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Set_Oinsert_001t__HOL__Obool,type,
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tff(sy_c_Set_Oinsert_001t__Int__Oint,type,
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tff(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Set_Oinsert_001t__Nat__Onat,type,
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tff(sy_c_Set_Oinsert_001t__Num__Onum,type,
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tff(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Set_Oinsert_001t__Rat__Orat,type,
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tff(sy_c_Set_Oinsert_001t__Real__Oreal,type,
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tff(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Set_Ois__empty_001t__HOL__Obool,type,
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tff(sy_c_Set_Ois__empty_001t__Int__Oint,type,
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tff(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
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tff(sy_c_Set_Ois__empty_001t__Real__Oreal,type,
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tff(sy_c_Set_Ois__singleton_001t__Complex__Ocomplex,type,
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tff(sy_c_Set_Ois__singleton_001t__HOL__Obool,type,
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tff(sy_c_Set_Ois__singleton_001t__Int__Oint,type,
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tff(sy_c_Set_Ois__singleton_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
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tff(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_Oremove_001t__HOL__Obool,type,
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tff(sy_c_Set_Oremove_001t__Int__Oint,type,
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tff(sy_c_Set_Oremove_001t__Nat__Onat,type,
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tff(sy_c_Set_Oremove_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_Othe__elem_001t__HOL__Obool,type,
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tff(sy_c_Set_Othe__elem_001t__Int__Oint,type,
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tff(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
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tff(sy_c_Set_Othe__elem_001t__Real__Oreal,type,
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tff(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
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tff(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
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tff(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
    topolo4267028734544971653eq_rat: fun_nat_rat > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
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tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
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tff(sy_c_Transcendental_Oarccos,type,
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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,
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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_Odiffs_001t__Complex__Ocomplex,type,
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tff(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
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tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
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tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    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,
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tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opowr__real,type,
    powr_real2: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
    sinh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
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tff(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
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tff(sy_c_Typedef_Otype__definition_001t__Product____Type__Ounit_001t__HOL__Obool,type,
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tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( bool * bool ) > vEBT_VEBT ).

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

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: fun_VE3763956218935105199t_bool ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr817965127378462947t_bool ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr817965127378462947t_bool ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr817965127378462947t_bool ).

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

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

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

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_bool ).

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

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H,type,
    vEBT_VEBT_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H__rel,type,
    vEBT_VEBT_insert_rel: fun_Pr817965127378462947t_bool ).

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

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr817965127378462947t_bool ).

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

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_bool ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun_VE5877677516038742509T_bool ).

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

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

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: fun_op7792301606653096457t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater__rel,type,
    vEBT_V5711637165310795180er_rel: fun_Pr1775001904083774893t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: fun_op7792301606653096457t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless__rel,type,
    vEBT_VEBT_less_rel: fun_Pr1775001904083774893t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: fun_op7792301606653096457t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq__rel,type,
    vEBT_VEBT_lesseq_rel: fun_Pr1775001904083774893t_bool ).

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

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__comp__shift_001t__Nat__Onat,type,
    vEBT_V2881884560877996034ft_nat: fun_nat_fun_nat_bool > fun_op7792301606653096457t_bool ).

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

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

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Nat__Onat,type,
    vEBT_V3895251965096974666el_nat: fun_Pr4582292742318232305t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Num__Onum,type,
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tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun_op7210469962420818650on_nat ).

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

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun_VE5877677516038742509T_bool ).

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

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun_VE5877677516038742509T_bool ).

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

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

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun_Pr817965127378462947t_bool ).

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

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

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun_Pr817965127378462947t_bool ).

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

tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: fun_nat_fun_nat_bool > fun_nat_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: ( fun_Pr817965127378462947t_bool * produc9072475918466114483BT_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
    accp_P5526603486790338531on_nat: ( fun_Pr4582292742318232305t_bool * produc7004362771917097900on_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Num__Onum_Mt__fun_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__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_Mt__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_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
    accp_VEBT_VEBT: fun_VE5877677516038742509T_bool > fun_VEBT_VEBT_bool ).

tff(sy_c_Wellfounded_Ofinite__psubset_001t__Complex__Ocomplex,type,
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tff(sy_c_Wellfounded_Ofinite__psubset_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Wellfounded_Ofinite__psubset_001t__Int__Oint,type,
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tff(sy_c_Wellfounded_Ofinite__psubset_001t__Nat__Onat,type,
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tff(sy_c_Wellfounded_Oless__than,type,
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tff(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Nat__Onat,type,
    lex_prod_nat_nat: ( set_Pr1261947904930325089at_nat * set_Pr1261947904930325089at_nat ) > set_Pr8693737435421807431at_nat ).

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tff(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
    measure_int: fun_int_nat > set_Pr958786334691620121nt_int ).

tff(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
    measure_nat: fun_nat_nat > set_Pr1261947904930325089at_nat ).

tff(sy_c_Wellfounded_Omeasure_001t__Option__Ooption_It__Nat__Onat_J,type,
    measure_option_nat: fun_option_nat_nat > set_Pr6588086440996610945on_nat ).

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tff(sy_c_Wellfounded_Omin__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
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tff(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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% Relevant facts (10745)
tff(fact_0__092_060open_062vebt__mint_At_A_092_060noteq_062_Avebt__mint_Ak_092_060close_062,axiom,
    vEBT_vebt_mint(ta) != vEBT_vebt_mint(k) ).

% \<open>vebt_mint t \<noteq> vebt_mint k\<close>
tff(fact_1_abdef,axiom,
    ( ( ( vEBT_vebt_mint(ta) = none_nat )
      & ( vEBT_vebt_mint(k) = some_nat(b) ) )
    | ( ( vEBT_vebt_mint(ta) = some_nat(a) )
      & ( vEBT_vebt_mint(k) = none_nat ) )
    | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,a),b))
      & ( some_nat(a) = vEBT_vebt_mint(ta) )
      & ( some_nat(b) = vEBT_vebt_mint(k) ) )
    | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,b),a))
      & ( some_nat(a) = vEBT_vebt_mint(ta) )
      & ( some_nat(b) = vEBT_vebt_mint(k) ) ) ) ).

% abdef
tff(fact_2_assms_I3_J,axiom,
    vEBT_VEBT_set_vebt(ta) = vEBT_VEBT_set_vebt(k) ).

% assms(3)
tff(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062a_Ab_O_Avebt__mint_At_A_061_ANone_A_092_060and_062_Avebt__mint_Ak_A_061_ASome_Ab_A_092_060or_062_Avebt__mint_At_A_061_ASome_Aa_A_092_060and_062_Avebt__mint_Ak_A_061_ANone_A_092_060or_062_Aa_A_060_Ab_A_092_060and_062_ASome_Aa_A_061_Avebt__mint_At_A_092_060and_062_ASome_Ab_A_061_Avebt__mint_Ak_A_092_060or_062_Ab_A_060_Aa_A_092_060and_062_ASome_Aa_A_061_Avebt__mint_At_A_092_060and_062_ASome_Ab_A_061_Avebt__mint_Ak_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [A: nat,B: nat] :
        ~ ( ( ( vEBT_vebt_mint(ta) = none_nat )
            & ( vEBT_vebt_mint(k) = some_nat(B) ) )
          | ( ( vEBT_vebt_mint(ta) = some_nat(A) )
            & ( vEBT_vebt_mint(k) = none_nat ) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
            & ( some_nat(A) = vEBT_vebt_mint(ta) )
            & ( some_nat(B) = vEBT_vebt_mint(k) ) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
            & ( some_nat(A) = vEBT_vebt_mint(ta) )
            & ( some_nat(B) = vEBT_vebt_mint(k) ) ) ) ).

% \<open>\<And>thesis. (\<And>a b. vebt_mint t = None \<and> vebt_mint k = Some b \<or> vebt_mint t = Some a \<and> vebt_mint k = None \<or> a < b \<and> Some a = vebt_mint t \<and> Some b = vebt_mint k \<or> b < a \<and> Some a = vebt_mint t \<and> Some b = vebt_mint k \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_4_greater__shift,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(X)),some_nat(Y))) ) ).

% greater_shift
tff(fact_5_less__shift,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(X)),some_nat(Y))) ) ).

% less_shift
tff(fact_6_not__None__eq,axiom,
    ! [X: option_nat] :
      ( ( X != none_nat )
    <=> ? [Y2: nat] : X = some_nat(Y2) ) ).

% not_None_eq
tff(fact_7_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
    <=> ? [Y2: product_prod_nat_nat] : X = some_P7363390416028606310at_nat(Y2) ) ).

% not_None_eq
tff(fact_8_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
    <=> ? [Y2: num] : X = some_num(Y2) ) ).

% not_None_eq
tff(fact_9_not__Some__eq,axiom,
    ! [X: option_nat] :
      ( ! [Y2: nat] : X != some_nat(Y2)
    <=> ( X = none_nat ) ) ).

% not_Some_eq
tff(fact_10_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ! [Y2: product_prod_nat_nat] : X != some_P7363390416028606310at_nat(Y2)
    <=> ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_11_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ! [Y2: num] : X != some_num(Y2)
    <=> ( X = none_num ) ) ).

% not_Some_eq
tff(fact_12_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ( vEBT_vebt_mint(T) = none_nat ) ) ).

% minNullmin
tff(fact_13_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(T) = none_nat )
     => pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T)) ) ).

% minminNull
tff(fact_14_option_Oinject,axiom,
    ! [X2: nat,Y22: nat] :
      ( ( some_nat(X2) = some_nat(Y22) )
    <=> ( X2 = Y22 ) ) ).

% option.inject
tff(fact_15_option_Oinject,axiom,
    ! [X2: product_prod_nat_nat,Y22: product_prod_nat_nat] :
      ( ( some_P7363390416028606310at_nat(X2) = some_P7363390416028606310at_nat(Y22) )
    <=> ( X2 = Y22 ) ) ).

% option.inject
tff(fact_16_option_Oinject,axiom,
    ! [X2: num,Y22: num] :
      ( ( some_num(X2) = some_num(Y22) )
    <=> ( X2 = Y22 ) ) ).

% option.inject
tff(fact_17_assms_I2_J,axiom,
    pp(aa_nat_bool(vEBT_invar_vebt(k),h)) ).

% assms(2)
tff(fact_18_assms_I1_J,axiom,
    pp(aa_nat_bool(vEBT_invar_vebt(ta),h)) ).

% assms(1)
tff(fact_19_option_Odistinct_I1_J,axiom,
    ! [X2: nat] : none_nat != some_nat(X2) ).

% option.distinct(1)
tff(fact_20_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X2) ).

% option.distinct(1)
tff(fact_21_option_Odistinct_I1_J,axiom,
    ! [X2: num] : none_num != some_num(X2) ).

% option.distinct(1)
tff(fact_22_option_OdiscI,axiom,
    ! [Option: option_nat,X2: nat] :
      ( ( Option = some_nat(X2) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_23_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X2) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_24_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option = some_num(X2) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_25_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X22: nat] : Y != some_nat(X22) ) ).

% option.exhaust
tff(fact_26_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X22: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X22) ) ).

% option.exhaust
tff(fact_27_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X22: num] : Y != some_num(X22) ) ).

% option.exhaust
tff(fact_28_insert_H__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_VEBT_insert(T,X)),N)) ) ).

% insert'_pres_valid
tff(fact_29_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),X))
       => ( vEBT_vebt_mint(T) = some_nat(X) ) ) ) ).

% mint_sound
tff(fact_30_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(X) )
       => pp(aa_nat_bool(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),X)) ) ) ).

% mint_corr
tff(fact_31_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_32_combine__options__cases,axiom,
    ! [X: option_nat,P: fun_op7792301606653096457t_bool,Y: option_nat] :
      ( ( ( X = none_nat )
       => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X),Y)) )
     => ( ( ( Y = none_nat )
         => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X),Y)) )
       => ( ! [A: nat,B: nat] :
              ( ( X = some_nat(A) )
             => ( ( Y = some_nat(B) )
               => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X),Y)) ) )
         => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_33_combine__options__cases,axiom,
    ! [X: option_nat,P: fun_op5698422665285572208t_bool,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_nat )
       => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X),Y)) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X),Y)) )
       => ( ! [A: nat,B: product_prod_nat_nat] :
              ( ( X = some_nat(A) )
             => ( ( Y = some_P7363390416028606310at_nat(B) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X),Y)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_34_combine__options__cases,axiom,
    ! [X: option_nat,P: fun_op8690601775248757823m_bool,Y: option_num] :
      ( ( ( X = none_nat )
       => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X),Y)) )
     => ( ( ( Y = none_num )
         => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X),Y)) )
       => ( ! [A: nat,B: num] :
              ( ( X = some_nat(A) )
             => ( ( Y = some_num(B) )
               => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X),Y)) ) )
         => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_35_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: fun_op9100744243108201484t_bool,Y: option_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X),Y)) )
     => ( ( ( Y = none_nat )
         => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X),Y)) )
       => ( ! [A: product_prod_nat_nat,B: nat] :
              ( ( X = some_P7363390416028606310at_nat(A) )
             => ( ( Y = some_nat(B) )
               => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X),Y)) ) )
         => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_36_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: fun_op5682630191948829869t_bool,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X),Y)) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X),Y)) )
       => ( ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
              ( ( X = some_P7363390416028606310at_nat(A) )
             => ( ( Y = some_P7363390416028606310at_nat(B) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X),Y)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_37_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: fun_op775672374849087042m_bool,Y: option_num] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X),Y)) )
     => ( ( ( Y = none_num )
         => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X),Y)) )
       => ( ! [A: product_prod_nat_nat,B: num] :
              ( ( X = some_P7363390416028606310at_nat(A) )
             => ( ( Y = some_num(B) )
               => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X),Y)) ) )
         => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_38_combine__options__cases,axiom,
    ! [X: option_num,P: fun_op1064277265382982207t_bool,Y: option_nat] :
      ( ( ( X = none_num )
       => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X),Y)) )
     => ( ( ( Y = none_nat )
         => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X),Y)) )
       => ( ! [A: num,B: nat] :
              ( ( X = some_num(A) )
             => ( ( Y = some_nat(B) )
               => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X),Y)) ) )
         => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_39_combine__options__cases,axiom,
    ! [X: option_num,P: fun_op7732454363416259258t_bool,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_num )
       => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X),Y)) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X),Y)) )
       => ( ! [A: num,B: product_prod_nat_nat] :
              ( ( X = some_num(A) )
             => ( ( Y = some_P7363390416028606310at_nat(B) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X),Y)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_40_combine__options__cases,axiom,
    ! [X: option_num,P: fun_op1962577433978643573m_bool,Y: option_num] :
      ( ( ( X = none_num )
       => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X),Y)) )
     => ( ( ( Y = none_num )
         => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X),Y)) )
       => ( ! [A: num,B: num] :
              ( ( X = some_num(A) )
             => ( ( Y = some_num(B) )
               => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X),Y)) ) )
         => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_41_split__option__all,axiom,
    ! [P: fun_option_nat_bool] :
      ( ! [X_1: option_nat] : pp(aa_option_nat_bool(P,X_1))
    <=> ( pp(aa_option_nat_bool(P,none_nat))
        & ! [X3: nat] : pp(aa_option_nat_bool(P,some_nat(X3))) ) ) ).

% split_option_all
tff(fact_42_split__option__all,axiom,
    ! [P: fun_op2627489687811228755t_bool] :
      ( ! [X_1: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P,X_1))
    <=> ( pp(aa_opt6800557945374683724t_bool(P,none_P5556105721700978146at_nat))
        & ! [X3: product_prod_nat_nat] : pp(aa_opt6800557945374683724t_bool(P,some_P7363390416028606310at_nat(X3))) ) ) ).

% split_option_all
tff(fact_43_split__option__all,axiom,
    ! [P: fun_option_num_bool] :
      ( ! [X_1: option_num] : pp(aa_option_num_bool(P,X_1))
    <=> ( pp(aa_option_num_bool(P,none_num))
        & ! [X3: num] : pp(aa_option_num_bool(P,some_num(X3))) ) ) ).

% split_option_all
tff(fact_44_split__option__ex,axiom,
    ! [P: fun_option_nat_bool] :
      ( ? [X_1: option_nat] : pp(aa_option_nat_bool(P,X_1))
    <=> ( pp(aa_option_nat_bool(P,none_nat))
        | ? [X3: nat] : pp(aa_option_nat_bool(P,some_nat(X3))) ) ) ).

% split_option_ex
tff(fact_45_split__option__ex,axiom,
    ! [P: fun_op2627489687811228755t_bool] :
      ( ? [X_1: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P,X_1))
    <=> ( pp(aa_opt6800557945374683724t_bool(P,none_P5556105721700978146at_nat))
        | ? [X3: product_prod_nat_nat] : pp(aa_opt6800557945374683724t_bool(P,some_P7363390416028606310at_nat(X3))) ) ) ).

% split_option_ex
tff(fact_46_split__option__ex,axiom,
    ! [P: fun_option_num_bool] :
      ( ? [X_1: option_num] : pp(aa_option_num_bool(P,X_1))
    <=> ( pp(aa_option_num_bool(P,none_num))
        | ? [X3: num] : pp(aa_option_num_bool(P,some_num(X3))) ) ) ).

% split_option_ex
tff(fact_47_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_mint(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_48_mint__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(Maxi) )
       => pp(aa_nat_bool(vEBT_vebt_member(T),Maxi)) ) ) ).

% mint_member
tff(fact_49_valid__eq,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(T),D))
    <=> pp(aa_nat_bool(vEBT_invar_vebt(T),D)) ) ).

% valid_eq
tff(fact_50_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),D))
     => pp(aa_nat_bool(vEBT_VEBT_valid(T),D)) ) ).

% valid_eq1
tff(fact_51_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(T),D))
     => pp(aa_nat_bool(vEBT_invar_vebt(T),D)) ) ).

% valid_eq2
tff(fact_52_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% deg_not_0
tff(fact_53_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_set_nat_bool(finite_finite_nat,vEBT_VEBT_set_vebt(T))) ) ).

% set_vebt_finite
tff(fact_54_case4_I12_J,axiom,
    pp(aa_nat_bool(vEBT_invar_vebt(sa),deg)) ).

% case4(12)
tff(fact_55_succ__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_succ(T,X) = some_nat(Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),X),Sx)) ) ) ).

% succ_corr
tff(fact_56_pred__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Px: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_pred(T,X) = some_nat(Px) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),X),Px)) ) ) ).

% pred_corr
tff(fact_57_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).

% valid_tree_deg_neq_0
tff(fact_58_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).

% valid_0_not
tff(fact_59_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ~ pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ).

% min_Null_member
tff(fact_60_pred__none__empty,axiom,
    ! [Xs: set_nat,A2: nat] :
      ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(Xs),A2),X_12))
     => ( pp(aa_set_nat_bool(finite_finite_nat,Xs))
       => ~ ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),Xs))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),A2)) ) ) ) ).

% pred_none_empty
tff(fact_61_succ__none__empty,axiom,
    ! [Xs: set_nat,A2: nat] :
      ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(Xs),A2),X_12))
     => ( pp(aa_set_nat_bool(finite_finite_nat,Xs))
       => ~ ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),Xs))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),X4)) ) ) ) ).

% succ_none_empty
tff(fact_62_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(vEBT_vebt_member(T),X))
      <=> pp(aa_set_nat_bool(member_nat(X),vEBT_set_vebt(T))) ) ) ).

% member_correct
tff(fact_63_obtain__set__pred,axiom,
    ! [Z: nat,X: nat,A3: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X))
     => ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(A3),Z))
       => ( pp(aa_set_nat_bool(finite_finite_nat,A3))
         => ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(A3),X),X_12)) ) ) ) ).

% obtain_set_pred
tff(fact_64_pred__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_pred(T,X) = some_nat(Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_set_vebt(T)),X),Sx)) ) ) ).

% pred_correct
tff(fact_65_mem__Collect__eq,axiom,
    ! [A2: bool,P: fun_bool_bool] :
      ( pp(aa_set_bool_bool(member_bool(A2),collect_bool(P)))
    <=> pp(aa_bool_bool(P,A2)) ) ).

% mem_Collect_eq
tff(fact_66_mem__Collect__eq,axiom,
    ! [A2: real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(member_real(A2),collect_real(P)))
    <=> pp(aa_real_bool(P,A2)) ) ).

% mem_Collect_eq
tff(fact_67_mem__Collect__eq,axiom,
    ! [A2: list_nat,P: fun_list_nat_bool] :
      ( member_list_nat(A2,collect_list_nat(P))
    <=> pp(aa_list_nat_bool(P,A2)) ) ).

% mem_Collect_eq
tff(fact_68_mem__Collect__eq,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A2),collect_set_nat(P)))
    <=> pp(aa_set_nat_bool(P,A2)) ) ).

% mem_Collect_eq
tff(fact_69_mem__Collect__eq,axiom,
    ! [A2: nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(member_nat(A2),collect_nat(P)))
    <=> pp(aa_nat_bool(P,A2)) ) ).

% mem_Collect_eq
tff(fact_70_mem__Collect__eq,axiom,
    ! [A2: int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(member_int(A2),collect_int(P)))
    <=> pp(aa_int_bool(P,A2)) ) ).

% mem_Collect_eq
tff(fact_71_Collect__mem__eq,axiom,
    ! [A3: set_bool] : collect_bool(aTP_Lamm_a(A3)) = A3 ).

% Collect_mem_eq
tff(fact_72_Collect__mem__eq,axiom,
    ! [A3: set_real] : collect_real(aTP_Lamm_aa(A3)) = A3 ).

% Collect_mem_eq
tff(fact_73_Collect__mem__eq,axiom,
    ! [A3: set_list_nat] : collect_list_nat(aTP_Lamm_ab(A3)) = A3 ).

% Collect_mem_eq
tff(fact_74_Collect__mem__eq,axiom,
    ! [A3: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A3)) = A3 ).

% Collect_mem_eq
tff(fact_75_Collect__mem__eq,axiom,
    ! [A3: set_nat] : collect_nat(aTP_Lamm_ad(A3)) = A3 ).

% Collect_mem_eq
tff(fact_76_Collect__mem__eq,axiom,
    ! [A3: set_int] : collect_int(aTP_Lamm_ae(A3)) = A3 ).

% Collect_mem_eq
tff(fact_77_Collect__cong,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( ! [X5: real] :
          ( pp(aa_real_bool(P,X5))
        <=> pp(aa_real_bool(Q,X5)) )
     => ( collect_real(P) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_78_Collect__cong,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ! [X5: list_nat] :
          ( pp(aa_list_nat_bool(P,X5))
        <=> pp(aa_list_nat_bool(Q,X5)) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_79_Collect__cong,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ! [X5: set_nat] :
          ( pp(aa_set_nat_bool(P,X5))
        <=> pp(aa_set_nat_bool(Q,X5)) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_80_Collect__cong,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X5: nat] :
          ( pp(aa_nat_bool(P,X5))
        <=> pp(aa_nat_bool(Q,X5)) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_81_Collect__cong,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X5: int] :
          ( pp(aa_int_bool(P,X5))
        <=> pp(aa_int_bool(Q,X5)) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_82_succ__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_succ(T,X) = some_nat(Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_set_vebt(T)),X),Sx)) ) ) ).

% succ_correct
tff(fact_83_obtain__set__succ,axiom,
    ! [X: nat,Z: nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z))
     => ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(A3),Z))
       => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
         => ( ( A3 = B2 )
           => ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(A3),X),X_12)) ) ) ) ) ).

% obtain_set_succ
tff(fact_84_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_buildup(N)),N)) ) ).

% buildup_gives_valid
tff(fact_85_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_86_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_87_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% neq0_conv
tff(fact_88_less__nat__zero__code,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_nat_zero_code
tff(fact_89_not__gr__zero,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_90_succ__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),X),Y))
    <=> ( pp(aa_nat_bool(vEBT_vebt_member(T),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        & ! [Z2: nat] :
            ( ( pp(aa_nat_bool(vEBT_vebt_member(T),Z2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z2)) ) ) ) ).

% succ_member
tff(fact_91_pred__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),X),Y))
    <=> ( pp(aa_nat_bool(vEBT_vebt_member(T),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
        & ! [Z2: nat] :
            ( ( pp(aa_nat_bool(vEBT_vebt_member(T),Z2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z2),Y)) ) ) ) ).

% pred_member
tff(fact_92_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(Mini) )
       => ( pp(aa_nat_bool(vEBT_vebt_member(T),X))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mini),X)) ) ) ) ).

% mint_corr_help
tff(fact_93_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_maxt(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_94_ex__min__if__finite,axiom,
    ! [S: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ( S != bot_bo7653980558646680370d_enat )
       => ? [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),S))
            & ~ ? [Xa: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),S))
                  & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Xa),X5)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_95_ex__min__if__finite,axiom,
    ! [S: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( ( S != bot_bot_set_bool )
       => ? [X5: bool] :
            ( pp(aa_set_bool_bool(member_bool(X5),S))
            & ~ ? [Xa: bool] :
                  ( pp(aa_set_bool_bool(member_bool(Xa),S))
                  & pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,Xa),X5)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_96_ex__min__if__finite,axiom,
    ! [S: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( S != bot_bot_set_real )
       => ? [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),S))
            & ~ ? [Xa: real] :
                  ( pp(aa_set_real_bool(member_real(Xa),S))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xa),X5)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_97_ex__min__if__finite,axiom,
    ! [S: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,S))
     => ( ( S != bot_bot_set_rat )
       => ? [X5: rat] :
            ( pp(aa_set_rat_bool(member_rat(X5),S))
            & ~ ? [Xa: rat] :
                  ( pp(aa_set_rat_bool(member_rat(Xa),S))
                  & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Xa),X5)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_98_ex__min__if__finite,axiom,
    ! [S: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,S))
     => ( ( S != bot_bot_set_num )
       => ? [X5: num] :
            ( pp(aa_set_num_bool(member_num(X5),S))
            & ~ ? [Xa: num] :
                  ( pp(aa_set_num_bool(member_num(Xa),S))
                  & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Xa),X5)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_99_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ( S != bot_bot_set_nat )
       => ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),S))
            & ~ ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),S))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),X5)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_100_ex__min__if__finite,axiom,
    ! [S: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ( S != bot_bot_set_int )
       => ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),S))
            & ~ ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),S))
                  & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa),X5)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_101_infinite__growing,axiom,
    ! [X6: set_Extended_enat] :
      ( ( X6 != bot_bo7653980558646680370d_enat )
     => ( ! [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),X6))
           => ? [Xa: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),X6))
                & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X5),Xa)) ) )
       => ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,X6)) ) ) ).

% infinite_growing
tff(fact_102_infinite__growing,axiom,
    ! [X6: set_bool] :
      ( ( X6 != bot_bot_set_bool )
     => ( ! [X5: bool] :
            ( pp(aa_set_bool_bool(member_bool(X5),X6))
           => ? [Xa: bool] :
                ( pp(aa_set_bool_bool(member_bool(Xa),X6))
                & pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,X5),Xa)) ) )
       => ~ pp(aa_set_bool_bool(finite_finite_bool,X6)) ) ) ).

% infinite_growing
tff(fact_103_infinite__growing,axiom,
    ! [X6: set_real] :
      ( ( X6 != bot_bot_set_real )
     => ( ! [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),X6))
           => ? [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),X6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Xa)) ) )
       => ~ pp(aa_set_real_bool(finite_finite_real,X6)) ) ) ).

% infinite_growing
tff(fact_104_infinite__growing,axiom,
    ! [X6: set_rat] :
      ( ( X6 != bot_bot_set_rat )
     => ( ! [X5: rat] :
            ( pp(aa_set_rat_bool(member_rat(X5),X6))
           => ? [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),X6))
                & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Xa)) ) )
       => ~ pp(aa_set_rat_bool(finite_finite_rat,X6)) ) ) ).

% infinite_growing
tff(fact_105_infinite__growing,axiom,
    ! [X6: set_num] :
      ( ( X6 != bot_bot_set_num )
     => ( ! [X5: num] :
            ( pp(aa_set_num_bool(member_num(X5),X6))
           => ? [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),X6))
                & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Xa)) ) )
       => ~ pp(aa_set_num_bool(finite_finite_num,X6)) ) ) ).

% infinite_growing
tff(fact_106_infinite__growing,axiom,
    ! [X6: set_nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),X6))
           => ? [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),X6))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Xa)) ) )
       => ~ pp(aa_set_nat_bool(finite_finite_nat,X6)) ) ) ).

% infinite_growing
tff(fact_107_infinite__growing,axiom,
    ! [X6: set_int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),X6))
           => ? [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),X6))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Xa)) ) )
       => ~ pp(aa_set_int_bool(finite_finite_int,X6)) ) ) ).

% infinite_growing
tff(fact_108_max__in__set__def,axiom,
    ! [Xs: set_nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(Xs),X))
    <=> ( pp(aa_set_nat_bool(member_nat(X),Xs))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),X)) ) ) ) ).

% max_in_set_def
tff(fact_109_min__in__set__def,axiom,
    ! [Xs: set_nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(Xs),X))
    <=> ( pp(aa_set_nat_bool(member_nat(X),Xs))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),X3)) ) ) ) ).

% min_in_set_def
tff(fact_110_maxt__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => pp(aa_nat_bool(vEBT_vebt_member(T),Maxi)) ) ) ).

% maxt_member
tff(fact_111_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => ( pp(aa_nat_bool(vEBT_vebt_member(T),X))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Maxi)) ) ) ) ).

% maxt_corr_help
tff(fact_112_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(X) )
       => pp(aa_nat_bool(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),X)) ) ) ).

% maxt_corr
tff(fact_113_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),X))
       => ( vEBT_vebt_maxt(T) = some_nat(X) ) ) ) ).

% maxt_sound
tff(fact_114_le__zero__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_115_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2)) ).

% bot_nat_0.extremum
tff(fact_116_le0,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% le0
tff(fact_117_lesseq__shift,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,some_nat(X)),some_nat(Y))) ) ).

% lesseq_shift
tff(fact_118_le__refl,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N)) ).

% le_refl
tff(fact_119_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ) ).

% le_trans
tff(fact_120_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% eq_imp_le
tff(fact_121_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( M = N ) ) ) ).

% le_antisym
tff(fact_122_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).

% nat_le_linear
tff(fact_123_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B3: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B3)) )
       => ? [X5: nat] :
            ( pp(aa_nat_bool(P,X5))
            & ! [Y4: nat] :
                ( pp(aa_nat_bool(P,Y4))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y4),X5)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_124_zero__le,axiom,
    ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X)) ).

% zero_le
tff(fact_125_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% less_eq_nat.simps(1)
tff(fact_126_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
    <=> ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_127_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
     => ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_128_le__0__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_129_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J2))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J))) ) ) ).

% less_mono_imp_le_mono
tff(fact_130_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( ( M != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% le_neq_implies_less
tff(fact_131_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_or_eq_imp_le
tff(fact_132_le__eq__less__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_133_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_imp_le_nat
tff(fact_134_nat__less__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & ( M != N ) ) ) ).

% nat_less_le
tff(fact_135_ex__least__nat__le,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
            & ! [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K2))
               => ~ pp(aa_nat_bool(P,I3)) )
            & pp(aa_nat_bool(P,K2)) ) ) ) ).

% ex_least_nat_le
tff(fact_136_zero__reorient,axiom,
    ! [X: literal] :
      ( ( zero_zero_literal = X )
    <=> ( X = zero_zero_literal ) ) ).

% zero_reorient
tff(fact_137_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
    <=> ( X = zero_zero_real ) ) ).

% zero_reorient
tff(fact_138_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
    <=> ( X = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_139_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_140_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
    <=> ( X = zero_zero_int ) ) ).

% zero_reorient
tff(fact_141_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neqE_nat
tff(fact_142_infinite__descent,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ~ pp(aa_nat_bool(P,N2))
         => ? [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
              & ~ pp(aa_nat_bool(P,M2)) ) )
     => pp(aa_nat_bool(P,N)) ) ).

% infinite_descent
tff(fact_143_nat__less__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
             => pp(aa_nat_bool(P,M2)) )
         => pp(aa_nat_bool(P,N2)) )
     => pp(aa_nat_bool(P,N)) ) ).

% nat_less_induct
tff(fact_144_less__irrefl__nat,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_irrefl_nat
tff(fact_145_less__not__refl3,axiom,
    ! [S2: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S2),T))
     => ( S2 != T ) ) ).

% less_not_refl3
tff(fact_146_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( M != N ) ) ).

% less_not_refl2
tff(fact_147_less__not__refl,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_not_refl
tff(fact_148_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% nat_neq_iff
tff(fact_149_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_150_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_151_not__less__zero,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less_zero
tff(fact_152_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr_zeroI
tff(fact_153_infinite__descent0,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( ~ pp(aa_nat_bool(P,N2))
             => ? [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
                  & ~ pp(aa_nat_bool(P,M2)) ) ) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% infinite_descent0
tff(fact_154_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_155_less__zeroE,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_zeroE
tff(fact_156_not__less0,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less0
tff(fact_157_not__gr0,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr0
tff(fact_158_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr0I
tff(fact_159_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat)) ).

% bot_nat_0.extremum_strict
tff(fact_160_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_leaf
tff(fact_161_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(Xs),X),Y))
    <=> ( pp(aa_set_nat_bool(member_nat(Y),Xs))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3)) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_162_is__pred__in__set__def,axiom,
    ! [Xs: set_nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(Xs),X),Y))
    <=> ( pp(aa_set_nat_bool(member_nat(Y),Xs))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),X))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y)) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_163_finite__has__maximal,axiom,
    ! [A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ? [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A3))
            & ! [Xa: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),A3))
               => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X5),Xa))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_164_finite__has__maximal,axiom,
    ! [A3: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ( A3 != bot_bot_set_real )
       => ? [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),A3))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),Xa))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_165_finite__has__maximal,axiom,
    ! [A3: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( ( A3 != bot_bot_set_bool )
       => ? [X5: bool] :
            ( pp(aa_set_bool_bool(member_bool(X5),A3))
            & ! [Xa: bool] :
                ( pp(aa_set_bool_bool(member_bool(Xa),A3))
               => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,X5),Xa))
                 => ( pp(X5)
                  <=> pp(Xa) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_166_finite__has__maximal,axiom,
    ! [A3: set_set_int] :
      ( finite6197958912794628473et_int(A3)
     => ( ( A3 != bot_bot_set_set_int )
       => ? [X5: set_int] :
            ( member_set_int(X5,A3)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A3)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(X5),Xa))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_167_finite__has__maximal,axiom,
    ! [A3: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A3))
     => ( ( A3 != bot_bot_set_rat )
       => ? [X5: rat] :
            ( pp(aa_set_rat_bool(member_rat(X5),A3))
            & ! [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),A3))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Xa))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_168_finite__has__maximal,axiom,
    ! [A3: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,A3))
     => ( ( A3 != bot_bot_set_num )
       => ? [X5: num] :
            ( pp(aa_set_num_bool(member_num(X5),A3))
            & ! [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),A3))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Xa))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_169_finite__has__maximal,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( A3 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A3))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A3))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Xa))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_170_finite__has__maximal,axiom,
    ! [A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( A3 != bot_bot_set_int )
       => ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A3))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A3))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),Xa))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_171_finite__has__minimal,axiom,
    ! [A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ? [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A3))
            & ! [Xa: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),A3))
               => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,Xa),X5))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_172_finite__has__minimal,axiom,
    ! [A3: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ( A3 != bot_bot_set_real )
       => ? [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),A3))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X5))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_173_finite__has__minimal,axiom,
    ! [A3: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( ( A3 != bot_bot_set_bool )
       => ? [X5: bool] :
            ( pp(aa_set_bool_bool(member_bool(X5),A3))
            & ! [Xa: bool] :
                ( pp(aa_set_bool_bool(member_bool(Xa),A3))
               => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,Xa),X5))
                 => ( pp(X5)
                  <=> pp(Xa) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_174_finite__has__minimal,axiom,
    ! [A3: set_set_int] :
      ( finite6197958912794628473et_int(A3)
     => ( ( A3 != bot_bot_set_set_int )
       => ? [X5: set_int] :
            ( member_set_int(X5,A3)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A3)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(Xa),X5))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_175_finite__has__minimal,axiom,
    ! [A3: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A3))
     => ( ( A3 != bot_bot_set_rat )
       => ? [X5: rat] :
            ( pp(aa_set_rat_bool(member_rat(X5),A3))
            & ! [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),A3))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa),X5))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_176_finite__has__minimal,axiom,
    ! [A3: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,A3))
     => ( ( A3 != bot_bot_set_num )
       => ? [X5: num] :
            ( pp(aa_set_num_bool(member_num(X5),A3))
            & ! [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),A3))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa),X5))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_177_finite__has__minimal,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( A3 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A3))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A3))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X5))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_178_finite__has__minimal,axiom,
    ! [A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( A3 != bot_bot_set_int )
       => ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A3))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A3))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X5))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_179_empty__iff,axiom,
    ! [C: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(C),bot_bot_set_set_nat)) ).

% empty_iff
tff(fact_180_empty__iff,axiom,
    ! [C: real] : ~ pp(aa_set_real_bool(member_real(C),bot_bot_set_real)) ).

% empty_iff
tff(fact_181_empty__iff,axiom,
    ! [C: bool] : ~ pp(aa_set_bool_bool(member_bool(C),bot_bot_set_bool)) ).

% empty_iff
tff(fact_182_empty__iff,axiom,
    ! [C: nat] : ~ pp(aa_set_nat_bool(member_nat(C),bot_bot_set_nat)) ).

% empty_iff
tff(fact_183_empty__iff,axiom,
    ! [C: int] : ~ pp(aa_set_int_bool(member_int(C),bot_bot_set_int)) ).

% empty_iff
tff(fact_184_all__not__in__conv,axiom,
    ! [A3: set_set_nat] :
      ( ! [X3: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(X3),A3))
    <=> ( A3 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
tff(fact_185_all__not__in__conv,axiom,
    ! [A3: set_real] :
      ( ! [X3: real] : ~ pp(aa_set_real_bool(member_real(X3),A3))
    <=> ( A3 = bot_bot_set_real ) ) ).

% all_not_in_conv
tff(fact_186_all__not__in__conv,axiom,
    ! [A3: set_bool] :
      ( ! [X3: bool] : ~ pp(aa_set_bool_bool(member_bool(X3),A3))
    <=> ( A3 = bot_bot_set_bool ) ) ).

% all_not_in_conv
tff(fact_187_all__not__in__conv,axiom,
    ! [A3: set_nat] :
      ( ! [X3: nat] : ~ pp(aa_set_nat_bool(member_nat(X3),A3))
    <=> ( A3 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_188_all__not__in__conv,axiom,
    ! [A3: set_int] :
      ( ! [X3: int] : ~ pp(aa_set_int_bool(member_int(X3),A3))
    <=> ( A3 = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_189_Collect__empty__eq,axiom,
    ! [P: fun_list_nat_bool] :
      ( ( collect_list_nat(P) = bot_bot_set_list_nat )
    <=> ! [X3: list_nat] : ~ pp(aa_list_nat_bool(P,X3)) ) ).

% Collect_empty_eq
tff(fact_190_Collect__empty__eq,axiom,
    ! [P: fun_set_nat_bool] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ! [X3: set_nat] : ~ pp(aa_set_nat_bool(P,X3)) ) ).

% Collect_empty_eq
tff(fact_191_Collect__empty__eq,axiom,
    ! [P: fun_real_bool] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ! [X3: real] : ~ pp(aa_real_bool(P,X3)) ) ).

% Collect_empty_eq
tff(fact_192_Collect__empty__eq,axiom,
    ! [P: fun_bool_bool] :
      ( ( collect_bool(P) = bot_bot_set_bool )
    <=> ! [X3: bool] : ~ pp(aa_bool_bool(P,X3)) ) ).

% Collect_empty_eq
tff(fact_193_Collect__empty__eq,axiom,
    ! [P: fun_nat_bool] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ! [X3: nat] : ~ pp(aa_nat_bool(P,X3)) ) ).

% Collect_empty_eq
tff(fact_194_Collect__empty__eq,axiom,
    ! [P: fun_int_bool] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ! [X3: int] : ~ pp(aa_int_bool(P,X3)) ) ).

% Collect_empty_eq
tff(fact_195_empty__Collect__eq,axiom,
    ! [P: fun_list_nat_bool] :
      ( ( bot_bot_set_list_nat = collect_list_nat(P) )
    <=> ! [X3: list_nat] : ~ pp(aa_list_nat_bool(P,X3)) ) ).

% empty_Collect_eq
tff(fact_196_empty__Collect__eq,axiom,
    ! [P: fun_set_nat_bool] :
      ( ( bot_bot_set_set_nat = collect_set_nat(P) )
    <=> ! [X3: set_nat] : ~ pp(aa_set_nat_bool(P,X3)) ) ).

% empty_Collect_eq
tff(fact_197_empty__Collect__eq,axiom,
    ! [P: fun_real_bool] :
      ( ( bot_bot_set_real = collect_real(P) )
    <=> ! [X3: real] : ~ pp(aa_real_bool(P,X3)) ) ).

% empty_Collect_eq
tff(fact_198_empty__Collect__eq,axiom,
    ! [P: fun_bool_bool] :
      ( ( bot_bot_set_bool = collect_bool(P) )
    <=> ! [X3: bool] : ~ pp(aa_bool_bool(P,X3)) ) ).

% empty_Collect_eq
tff(fact_199_empty__Collect__eq,axiom,
    ! [P: fun_nat_bool] :
      ( ( bot_bot_set_nat = collect_nat(P) )
    <=> ! [X3: nat] : ~ pp(aa_nat_bool(P,X3)) ) ).

% empty_Collect_eq
tff(fact_200_empty__Collect__eq,axiom,
    ! [P: fun_int_bool] :
      ( ( bot_bot_set_int = collect_int(P) )
    <=> ! [X3: int] : ~ pp(aa_int_bool(P,X3)) ) ).

% empty_Collect_eq
tff(fact_201_order__refl,axiom,
    ! [X: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(X),X)) ).

% order_refl
tff(fact_202_order__refl,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),X)) ).

% order_refl
tff(fact_203_order__refl,axiom,
    ! [X: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),X)) ).

% order_refl
tff(fact_204_order__refl,axiom,
    ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),X)) ).

% order_refl
tff(fact_205_order__refl,axiom,
    ! [X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),X)) ).

% order_refl
tff(fact_206_dual__order_Orefl,axiom,
    ! [A2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A2),A2)) ).

% dual_order.refl
tff(fact_207_dual__order_Orefl,axiom,
    ! [A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),A2)) ).

% dual_order.refl
tff(fact_208_dual__order_Orefl,axiom,
    ! [A2: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),A2)) ).

% dual_order.refl
tff(fact_209_dual__order_Orefl,axiom,
    ! [A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),A2)) ).

% dual_order.refl
tff(fact_210_dual__order_Orefl,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),A2)) ).

% dual_order.refl
tff(fact_211_empty__subsetI,axiom,
    ! [A3: set_real] : pp(aa_set_real_bool(ord_less_eq_set_real(bot_bot_set_real),A3)) ).

% empty_subsetI
tff(fact_212_empty__subsetI,axiom,
    ! [A3: set_bool] : pp(aa_set_bool_bool(ord_less_eq_set_bool(bot_bot_set_bool),A3)) ).

% empty_subsetI
tff(fact_213_empty__subsetI,axiom,
    ! [A3: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,bot_bot_set_nat),A3)) ).

% empty_subsetI
tff(fact_214_empty__subsetI,axiom,
    ! [A3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(bot_bot_set_int),A3)) ).

% empty_subsetI
tff(fact_215_subset__empty,axiom,
    ! [A3: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),bot_bot_set_real))
    <=> ( A3 = bot_bot_set_real ) ) ).

% subset_empty
tff(fact_216_subset__empty,axiom,
    ! [A3: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),bot_bot_set_bool))
    <=> ( A3 = bot_bot_set_bool ) ) ).

% subset_empty
tff(fact_217_subset__empty,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),bot_bot_set_nat))
    <=> ( A3 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_218_subset__empty,axiom,
    ! [A3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),bot_bot_set_int))
    <=> ( A3 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_219_bot__set__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(bot_bo6469247547558333820t_bool) ).

% bot_set_def
tff(fact_220_bot__set__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(bot_bo8343409700046576086t_bool) ).

% bot_set_def
tff(fact_221_bot__set__def,axiom,
    bot_bot_set_real = collect_real(bot_bo4181092155253064880l_bool) ).

% bot_set_def
tff(fact_222_bot__set__def,axiom,
    bot_bot_set_bool = collect_bool(bot_bo4756488196683355415l_bool) ).

% bot_set_def
tff(fact_223_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_bool) ).

% bot_set_def
tff(fact_224_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_bool) ).

% bot_set_def
tff(fact_225_rev__finite__subset,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
       => pp(aa_set_nat_bool(finite_finite_nat,A3)) ) ) ).

% rev_finite_subset
tff(fact_226_rev__finite__subset,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),B2))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,A3)) ) ) ).

% rev_finite_subset
tff(fact_227_rev__finite__subset,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),B2))
       => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3)) ) ) ).

% rev_finite_subset
tff(fact_228_rev__finite__subset,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),B2))
       => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3)) ) ) ).

% rev_finite_subset
tff(fact_229_rev__finite__subset,axiom,
    ! [B2: set_int,A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
       => pp(aa_set_int_bool(finite_finite_int,A3)) ) ) ).

% rev_finite_subset
tff(fact_230_infinite__super,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S),T2))
     => ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
       => ~ pp(aa_set_nat_bool(finite_finite_nat,T2)) ) ) ).

% infinite_super
tff(fact_231_infinite__super,axiom,
    ! [S: set_complex,T2: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T2))
     => ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,T2)) ) ) ).

% infinite_super
tff(fact_232_infinite__super,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(S),T2))
     => ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S))
       => ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,T2)) ) ) ).

% infinite_super
tff(fact_233_infinite__super,axiom,
    ! [S: set_Extended_enat,T2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S),T2))
     => ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
       => ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T2)) ) ) ).

% infinite_super
tff(fact_234_infinite__super,axiom,
    ! [S: set_int,T2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S),T2))
     => ( ~ pp(aa_set_int_bool(finite_finite_int,S))
       => ~ pp(aa_set_int_bool(finite_finite_int,T2)) ) ) ).

% infinite_super
tff(fact_235_finite__subset,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => pp(aa_set_nat_bool(finite_finite_nat,A3)) ) ) ).

% finite_subset
tff(fact_236_finite__subset,axiom,
    ! [A3: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),B2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,A3)) ) ) ).

% finite_subset
tff(fact_237_finite__subset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),B2))
     => ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
       => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3)) ) ) ).

% finite_subset
tff(fact_238_finite__subset,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),B2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3)) ) ) ).

% finite_subset
tff(fact_239_finite__subset,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => pp(aa_set_int_bool(finite_finite_int,A3)) ) ) ).

% finite_subset
tff(fact_240_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_241_order__antisym__conv,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_242_order__antisym__conv,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_243_order__antisym__conv,axiom,
    ! [Y: num,X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_244_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_245_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_246_linorder__le__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% linorder_le_cases
tff(fact_247_linorder__le__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% linorder_le_cases
tff(fact_248_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_le_cases
tff(fact_249_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_le_cases
tff(fact_250_ord__le__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( ( aa_rat_rat(F,B3) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_251_ord__le__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( ( aa_rat_num(F,B3) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_252_ord__le__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( ( aa_rat_nat(F,B3) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_253_ord__le__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( ( aa_rat_int(F,B3) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X5)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_254_ord__le__eq__subst,axiom,
    ! [A2: num,B3: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( ( aa_num_rat(F,B3) = C )
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_255_ord__le__eq__subst,axiom,
    ! [A2: num,B3: num,F: fun_num_num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( ( aa_num_num(F,B3) = C )
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X5)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_256_ord__le__eq__subst,axiom,
    ! [A2: num,B3: num,F: fun_num_nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( ( aa_num_nat(F,B3) = C )
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X5)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_257_ord__le__eq__subst,axiom,
    ! [A2: num,B3: num,F: fun_num_int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( ( aa_num_int(F,B3) = C )
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X5)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_258_ord__le__eq__subst,axiom,
    ! [A2: nat,B3: nat,F: fun_nat_rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( ( aa_nat_rat(F,B3) = C )
       => ( ! [X5: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X5)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_259_ord__le__eq__subst,axiom,
    ! [A2: nat,B3: nat,F: fun_nat_num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( ( aa_nat_num(F,B3) = C )
       => ( ! [X5: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X5)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_260_ord__eq__le__subst,axiom,
    ! [A2: rat,F: fun_rat_rat,B3: rat,C: rat] :
      ( ( A2 = aa_rat_rat(F,B3) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_261_ord__eq__le__subst,axiom,
    ! [A2: num,F: fun_rat_num,B3: rat,C: rat] :
      ( ( A2 = aa_rat_num(F,B3) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_rat_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_262_ord__eq__le__subst,axiom,
    ! [A2: nat,F: fun_rat_nat,B3: rat,C: rat] :
      ( ( A2 = aa_rat_nat(F,B3) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_rat_nat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_263_ord__eq__le__subst,axiom,
    ! [A2: int,F: fun_rat_int,B3: rat,C: rat] :
      ( ( A2 = aa_rat_int(F,B3) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X5)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_rat_int(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_264_ord__eq__le__subst,axiom,
    ! [A2: rat,F: fun_num_rat,B3: num,C: num] :
      ( ( A2 = aa_num_rat(F,B3) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_num_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_265_ord__eq__le__subst,axiom,
    ! [A2: num,F: fun_num_num,B3: num,C: num] :
      ( ( A2 = aa_num_num(F,B3) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X5)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_num_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_266_ord__eq__le__subst,axiom,
    ! [A2: nat,F: fun_num_nat,B3: num,C: num] :
      ( ( A2 = aa_num_nat(F,B3) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X5)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_num_nat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_267_ord__eq__le__subst,axiom,
    ! [A2: int,F: fun_num_int,B3: num,C: num] :
      ( ( A2 = aa_num_int(F,B3) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X5)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_num_int(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_268_ord__eq__le__subst,axiom,
    ! [A2: rat,F: fun_nat_rat,B3: nat,C: nat] :
      ( ( A2 = aa_nat_rat(F,B3) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),C))
       => ( ! [X5: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X5)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_nat_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_269_ord__eq__le__subst,axiom,
    ! [A2: num,F: fun_nat_num,B3: nat,C: nat] :
      ( ( A2 = aa_nat_num(F,B3) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),C))
       => ( ! [X5: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X5)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_nat_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_270_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% linorder_linear
tff(fact_271_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% linorder_linear
tff(fact_272_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_linear
tff(fact_273_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_linear
tff(fact_274_order__eq__refl,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
     => pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).

% order_eq_refl
tff(fact_275_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% order_eq_refl
tff(fact_276_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% order_eq_refl
tff(fact_277_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% order_eq_refl
tff(fact_278_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% order_eq_refl
tff(fact_279_order__subst2,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,B3)),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_280_order__subst2,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,B3)),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_281_order__subst2,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,B3)),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_282_order__subst2,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,B3)),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X5)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_283_order__subst2,axiom,
    ! [A2: num,B3: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,B3)),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_284_order__subst2,axiom,
    ! [A2: num,B3: num,F: fun_num_num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,B3)),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X5)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_285_order__subst2,axiom,
    ! [A2: num,B3: num,F: fun_num_nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,B3)),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X5)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_286_order__subst2,axiom,
    ! [A2: num,B3: num,F: fun_num_int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,B3)),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X5)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_287_order__subst2,axiom,
    ! [A2: nat,B3: nat,F: fun_nat_rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,B3)),C))
       => ( ! [X5: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X5)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_288_order__subst2,axiom,
    ! [A2: nat,B3: nat,F: fun_nat_num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,B3)),C))
       => ( ! [X5: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X5)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_289_order__subst1,axiom,
    ! [A2: rat,F: fun_rat_rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(F,B3)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_290_order__subst1,axiom,
    ! [A2: rat,F: fun_num_rat,B3: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_num_rat(F,B3)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_num_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_291_order__subst1,axiom,
    ! [A2: rat,F: fun_nat_rat,B3: nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_nat_rat(F,B3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),C))
       => ( ! [X5: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X5)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_nat_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_292_order__subst1,axiom,
    ! [A2: rat,F: fun_int_rat,B3: int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_int_rat(F,B3)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),C))
       => ( ! [X5: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X5)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_int_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_293_order__subst1,axiom,
    ! [A2: num,F: fun_rat_num,B3: rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_rat_num(F,B3)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_rat_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_294_order__subst1,axiom,
    ! [A2: num,F: fun_num_num,B3: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_num_num(F,B3)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X5)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_num_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_295_order__subst1,axiom,
    ! [A2: num,F: fun_nat_num,B3: nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_nat_num(F,B3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),C))
       => ( ! [X5: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X5)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_nat_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_296_order__subst1,axiom,
    ! [A2: num,F: fun_int_num,B3: int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_int_num(F,B3)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),C))
       => ( ! [X5: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,X5)),aa_int_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_int_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_297_order__subst1,axiom,
    ! [A2: nat,F: fun_rat_nat,B3: rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_rat_nat(F,B3)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_rat_nat(F,C))) ) ) ) ).

% order_subst1
tff(fact_298_order__subst1,axiom,
    ! [A2: nat,F: fun_num_nat,B3: num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_num_nat(F,B3)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X5)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_num_nat(F,C))) ) ) ) ).

% order_subst1
tff(fact_299_Orderings_Oorder__eq__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
        & pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2)) ) ) ).

% Orderings.order_eq_iff
tff(fact_300_Orderings_Oorder__eq__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 = B3 )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2)) ) ) ).

% Orderings.order_eq_iff
tff(fact_301_Orderings_Oorder__eq__iff,axiom,
    ! [A2: num,B3: num] :
      ( ( A2 = B3 )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2)) ) ) ).

% Orderings.order_eq_iff
tff(fact_302_Orderings_Oorder__eq__iff,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 = B3 )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2)) ) ) ).

% Orderings.order_eq_iff
tff(fact_303_Orderings_Oorder__eq__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 = B3 )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2)) ) ) ).

% Orderings.order_eq_iff
tff(fact_304_antisym,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
       => ( A2 = B3 ) ) ) ).

% antisym
tff(fact_305_antisym,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
       => ( A2 = B3 ) ) ) ).

% antisym
tff(fact_306_antisym,axiom,
    ! [A2: num,B3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2))
       => ( A2 = B3 ) ) ) ).

% antisym
tff(fact_307_antisym,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
       => ( A2 = B3 ) ) ) ).

% antisym
tff(fact_308_antisym,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
       => ( A2 = B3 ) ) ) ).

% antisym
tff(fact_309_dual__order_Otrans,axiom,
    ! [B3: set_int,A2: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(C),B3))
       => pp(aa_set_int_bool(ord_less_eq_set_int(C),A2)) ) ) ).

% dual_order.trans
tff(fact_310_dual__order_Otrans,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A2)) ) ) ).

% dual_order.trans
tff(fact_311_dual__order_Otrans,axiom,
    ! [B3: num,A2: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A2)) ) ) ).

% dual_order.trans
tff(fact_312_dual__order_Otrans,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2)) ) ) ).

% dual_order.trans
tff(fact_313_dual__order_Otrans,axiom,
    ! [B3: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2)) ) ) ).

% dual_order.trans
tff(fact_314_dual__order_Oantisym,axiom,
    ! [B3: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
       => ( A2 = B3 ) ) ) ).

% dual_order.antisym
tff(fact_315_dual__order_Oantisym,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
       => ( A2 = B3 ) ) ) ).

% dual_order.antisym
tff(fact_316_dual__order_Oantisym,axiom,
    ! [B3: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
       => ( A2 = B3 ) ) ) ).

% dual_order.antisym
tff(fact_317_dual__order_Oantisym,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
       => ( A2 = B3 ) ) ) ).

% dual_order.antisym
tff(fact_318_dual__order_Oantisym,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
       => ( A2 = B3 ) ) ) ).

% dual_order.antisym
tff(fact_319_dual__order_Oeq__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
        & pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3)) ) ) ).

% dual_order.eq_iff
tff(fact_320_dual__order_Oeq__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 = B3 )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ) ).

% dual_order.eq_iff
tff(fact_321_dual__order_Oeq__iff,axiom,
    ! [A2: num,B3: num] :
      ( ( A2 = B3 )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3)) ) ) ).

% dual_order.eq_iff
tff(fact_322_dual__order_Oeq__iff,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 = B3 )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3)) ) ) ).

% dual_order.eq_iff
tff(fact_323_dual__order_Oeq__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 = B3 )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ) ).

% dual_order.eq_iff
tff(fact_324_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,A2: rat,B3: rat] :
      ( ! [A: rat,B: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A),B)) )
     => ( ! [A: rat,B: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(P,B),A))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A),B)) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A2),B3)) ) ) ).

% linorder_wlog
tff(fact_325_linorder__wlog,axiom,
    ! [P: fun_num_fun_num_bool,A2: num,B3: num] :
      ( ! [A: num,B: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A),B)) )
     => ( ! [A: num,B: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(P,B),A))
           => pp(aa_num_bool(aa_num_fun_num_bool(P,A),B)) )
       => pp(aa_num_bool(aa_num_fun_num_bool(P,A2),B3)) ) ) ).

% linorder_wlog
tff(fact_326_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_bool,A2: nat,B3: nat] :
      ( ! [A: nat,B: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) )
     => ( ! [A: nat,B: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B),A))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),B3)) ) ) ).

% linorder_wlog
tff(fact_327_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_bool,A2: int,B3: int] :
      ( ! [A: int,B: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A),B)) )
     => ( ! [A: int,B: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(P,B),A))
           => pp(aa_int_bool(aa_int_fun_int_bool(P,A),B)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A2),B3)) ) ) ).

% linorder_wlog
tff(fact_328_order__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),Z))
       => pp(aa_set_int_bool(ord_less_eq_set_int(X),Z)) ) ) ).

% order_trans
tff(fact_329_order__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z)) ) ) ).

% order_trans
tff(fact_330_order__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z)) ) ) ).

% order_trans
tff(fact_331_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z)) ) ) ).

% order_trans
tff(fact_332_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z)) ) ) ).

% order_trans
tff(fact_333_order_Otrans,axiom,
    ! [A2: set_int,B3: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),C))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A2),C)) ) ) ).

% order.trans
tff(fact_334_order_Otrans,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),C)) ) ) ).

% order.trans
tff(fact_335_order_Otrans,axiom,
    ! [A2: num,B3: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),C)) ) ) ).

% order.trans
tff(fact_336_order_Otrans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C)) ) ) ).

% order.trans
tff(fact_337_order_Otrans,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C)) ) ) ).

% order.trans
tff(fact_338_order__antisym,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_339_order__antisym,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_340_order__antisym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_341_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_342_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_343_ord__le__eq__trans,axiom,
    ! [A2: set_int,B3: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( ( B3 = C )
       => pp(aa_set_int_bool(ord_less_eq_set_int(A2),C)) ) ) ).

% ord_le_eq_trans
tff(fact_344_ord__le__eq__trans,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( ( B3 = C )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),C)) ) ) ).

% ord_le_eq_trans
tff(fact_345_ord__le__eq__trans,axiom,
    ! [A2: num,B3: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( ( B3 = C )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),C)) ) ) ).

% ord_le_eq_trans
tff(fact_346_ord__le__eq__trans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( ( B3 = C )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C)) ) ) ).

% ord_le_eq_trans
tff(fact_347_ord__le__eq__trans,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( ( B3 = C )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C)) ) ) ).

% ord_le_eq_trans
tff(fact_348_ord__eq__le__trans,axiom,
    ! [A2: set_int,B3: set_int,C: set_int] :
      ( ( A2 = B3 )
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),C))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A2),C)) ) ) ).

% ord_eq_le_trans
tff(fact_349_ord__eq__le__trans,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( A2 = B3 )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),C)) ) ) ).

% ord_eq_le_trans
tff(fact_350_ord__eq__le__trans,axiom,
    ! [A2: num,B3: num,C: num] :
      ( ( A2 = B3 )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),C)) ) ) ).

% ord_eq_le_trans
tff(fact_351_ord__eq__le__trans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( A2 = B3 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C)) ) ) ).

% ord_eq_le_trans
tff(fact_352_ord__eq__le__trans,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( A2 = B3 )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C)) ) ) ).

% ord_eq_le_trans
tff(fact_353_order__class_Oorder__eq__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
        & pp(aa_set_int_bool(ord_less_eq_set_int(Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_354_order__class_Oorder__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_355_order__class_Oorder__eq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_356_order__class_Oorder__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_357_order__class_Oorder__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_358_le__cases3,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
       => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) )
     => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z)) )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z))
           => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y)) )
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y))
             => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) )
           => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z))
               => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X)) )
             => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X))
                 => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_359_le__cases3,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
       => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z)) )
     => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z)) )
       => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z))
           => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y)) )
         => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y))
             => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) )
           => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
               => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X)) )
             => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X))
                 => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_360_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z)) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z)) )
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z))
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y)) )
         => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y))
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) )
           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
               => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X)) )
             => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X))
                 => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_361_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
       => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z)) )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z)) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y)) )
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y))
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) )
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
               => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X)) )
             => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X))
                 => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_362_nle__le,axiom,
    ! [A2: rat,B3: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
        & ( B3 != A2 ) ) ) ).

% nle_le
tff(fact_363_nle__le,axiom,
    ! [A2: num,B3: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2))
        & ( B3 != A2 ) ) ) ).

% nle_le
tff(fact_364_nle__le,axiom,
    ! [A2: nat,B3: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
        & ( B3 != A2 ) ) ) ).

% nle_le
tff(fact_365_nle__le,axiom,
    ! [A2: int,B3: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
        & ( B3 != A2 ) ) ) ).

% nle_le
tff(fact_366_lt__ex,axiom,
    ! [X: real] :
    ? [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X)) ).

% lt_ex
tff(fact_367_lt__ex,axiom,
    ! [X: rat] :
    ? [Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X)) ).

% lt_ex
tff(fact_368_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X)) ).

% lt_ex
tff(fact_369_gt__ex,axiom,
    ! [X: real] :
    ? [X_12: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X_12)) ).

% gt_ex
tff(fact_370_gt__ex,axiom,
    ! [X: rat] :
    ? [X_12: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),X_12)) ).

% gt_ex
tff(fact_371_gt__ex,axiom,
    ! [X: nat] :
    ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X_12)) ).

% gt_ex
tff(fact_372_gt__ex,axiom,
    ! [X: int] :
    ? [X_12: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X_12)) ).

% gt_ex
tff(fact_373_dense,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [Z3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z3))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),Y)) ) ) ).

% dense
tff(fact_374_dense,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ? [Z3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z3))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),Y)) ) ) ).

% dense
tff(fact_375_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_376_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_377_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_378_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_379_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_380_order_Oasym,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2)) ) ).

% order.asym
tff(fact_381_order_Oasym,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2)) ) ).

% order.asym
tff(fact_382_order_Oasym,axiom,
    ! [A2: num,B3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),A2)) ) ).

% order.asym
tff(fact_383_order_Oasym,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2)) ) ).

% order.asym
tff(fact_384_order_Oasym,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2)) ) ).

% order.asym
tff(fact_385_ord__eq__less__trans,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( A2 = B3 )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C)) ) ) ).

% ord_eq_less_trans
tff(fact_386_ord__eq__less__trans,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( A2 = B3 )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),C)) ) ) ).

% ord_eq_less_trans
tff(fact_387_ord__eq__less__trans,axiom,
    ! [A2: num,B3: num,C: num] :
      ( ( A2 = B3 )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),C)) ) ) ).

% ord_eq_less_trans
tff(fact_388_ord__eq__less__trans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( A2 = B3 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C)) ) ) ).

% ord_eq_less_trans
tff(fact_389_ord__eq__less__trans,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( A2 = B3 )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C)) ) ) ).

% ord_eq_less_trans
tff(fact_390_ord__less__eq__trans,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( ( B3 = C )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C)) ) ) ).

% ord_less_eq_trans
tff(fact_391_ord__less__eq__trans,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( ( B3 = C )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),C)) ) ) ).

% ord_less_eq_trans
tff(fact_392_ord__less__eq__trans,axiom,
    ! [A2: num,B3: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B3))
     => ( ( B3 = C )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),C)) ) ) ).

% ord_less_eq_trans
tff(fact_393_ord__less__eq__trans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( ( B3 = C )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C)) ) ) ).

% ord_less_eq_trans
tff(fact_394_ord__less__eq__trans,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( ( B3 = C )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C)) ) ) ).

% ord_less_eq_trans
tff(fact_395_less__induct,axiom,
    ! [P: fun_nat_bool,A2: nat] :
      ( ! [X5: nat] :
          ( ! [Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),X5))
             => pp(aa_nat_bool(P,Y4)) )
         => pp(aa_nat_bool(P,X5)) )
     => pp(aa_nat_bool(P,A2)) ) ).

% less_induct
tff(fact_396_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_397_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_398_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_399_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_400_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_401_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( ( X != Y )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_cases
tff(fact_402_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( ( X != Y )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_cases
tff(fact_403_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( ( X != Y )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_cases
tff(fact_404_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( ( X != Y )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_cases
tff(fact_405_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( ( X != Y )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_cases
tff(fact_406_dual__order_Oasym,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) ) ).

% dual_order.asym
tff(fact_407_dual__order_Oasym,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) ) ).

% dual_order.asym
tff(fact_408_dual__order_Oasym,axiom,
    ! [B3: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),A2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B3)) ) ).

% dual_order.asym
tff(fact_409_dual__order_Oasym,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3)) ) ).

% dual_order.asym
tff(fact_410_dual__order_Oasym,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3)) ) ).

% dual_order.asym
tff(fact_411_dual__order_Oirrefl,axiom,
    ! [A2: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),A2)) ).

% dual_order.irrefl
tff(fact_412_dual__order_Oirrefl,axiom,
    ! [A2: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),A2)) ).

% dual_order.irrefl
tff(fact_413_dual__order_Oirrefl,axiom,
    ! [A2: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),A2)) ).

% dual_order.irrefl
tff(fact_414_dual__order_Oirrefl,axiom,
    ! [A2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),A2)) ).

% dual_order.irrefl
tff(fact_415_dual__order_Oirrefl,axiom,
    ! [A2: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),A2)) ).

% dual_order.irrefl
tff(fact_416_exists__least__iff,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_1: nat] : pp(aa_nat_bool(P,X_1))
    <=> ? [N3: nat] :
          ( pp(aa_nat_bool(P,N3))
          & ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N3))
             => ~ pp(aa_nat_bool(P,M3)) ) ) ) ).

% exists_least_iff
tff(fact_417_linorder__less__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,A2: real,B3: real] :
      ( ! [A: real,B: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B)) )
     => ( ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),A))
       => ( ! [A: real,B: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B),A))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A2),B3)) ) ) ) ).

% linorder_less_wlog
tff(fact_418_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,A2: rat,B3: rat] :
      ( ! [A: rat,B: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A),B)) )
     => ( ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A),A))
       => ( ! [A: rat,B: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(P,B),A))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A),B)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A2),B3)) ) ) ) ).

% linorder_less_wlog
tff(fact_419_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_bool,A2: num,B3: num] :
      ( ! [A: num,B: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A),B)) )
     => ( ! [A: num] : pp(aa_num_bool(aa_num_fun_num_bool(P,A),A))
       => ( ! [A: num,B: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(P,B),A))
             => pp(aa_num_bool(aa_num_fun_num_bool(P,A),B)) )
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A2),B3)) ) ) ) ).

% linorder_less_wlog
tff(fact_420_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_bool,A2: nat,B3: nat] :
      ( ! [A: nat,B: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) )
     => ( ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),A))
       => ( ! [A: nat,B: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B),A))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),B3)) ) ) ) ).

% linorder_less_wlog
tff(fact_421_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_bool,A2: int,B3: int] :
      ( ! [A: int,B: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A),B)) )
     => ( ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(P,A),A))
       => ( ! [A: int,B: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(P,B),A))
             => pp(aa_int_bool(aa_int_fun_int_bool(P,A),B)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A2),B3)) ) ) ) ).

% linorder_less_wlog
tff(fact_422_order_Ostrict__trans,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C)) ) ) ).

% order.strict_trans
tff(fact_423_order_Ostrict__trans,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),C)) ) ) ).

% order.strict_trans
tff(fact_424_order_Ostrict__trans,axiom,
    ! [A2: num,B3: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),C)) ) ) ).

% order.strict_trans
tff(fact_425_order_Ostrict__trans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C)) ) ) ).

% order.strict_trans
tff(fact_426_order_Ostrict__trans,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C)) ) ) ).

% order.strict_trans
tff(fact_427_not__less__iff__gr__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_428_not__less__iff__gr__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_429_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_430_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_431_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_432_dual__order_Ostrict__trans,axiom,
    ! [B3: real,A2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A2)) ) ) ).

% dual_order.strict_trans
tff(fact_433_dual__order_Ostrict__trans,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A2)) ) ) ).

% dual_order.strict_trans
tff(fact_434_dual__order_Ostrict__trans,axiom,
    ! [B3: num,A2: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),A2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A2)) ) ) ).

% dual_order.strict_trans
tff(fact_435_dual__order_Ostrict__trans,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A2)) ) ) ).

% dual_order.strict_trans
tff(fact_436_dual__order_Ostrict__trans,axiom,
    ! [B3: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A2)) ) ) ).

% dual_order.strict_trans
tff(fact_437_order_Ostrict__implies__not__eq,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( A2 != B3 ) ) ).

% order.strict_implies_not_eq
tff(fact_438_order_Ostrict__implies__not__eq,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( A2 != B3 ) ) ).

% order.strict_implies_not_eq
tff(fact_439_order_Ostrict__implies__not__eq,axiom,
    ! [A2: num,B3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B3))
     => ( A2 != B3 ) ) ).

% order.strict_implies_not_eq
tff(fact_440_order_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( A2 != B3 ) ) ).

% order.strict_implies_not_eq
tff(fact_441_order_Ostrict__implies__not__eq,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( A2 != B3 ) ) ).

% order.strict_implies_not_eq
tff(fact_442_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
     => ( A2 != B3 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_443_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
     => ( A2 != B3 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_444_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B3: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),A2))
     => ( A2 != B3 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_445_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
     => ( A2 != B3 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_446_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
     => ( A2 != B3 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_447_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neqE
tff(fact_448_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_neqE
tff(fact_449_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_neqE
tff(fact_450_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neqE
tff(fact_451_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neqE
tff(fact_452_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_asym
tff(fact_453_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% order_less_asym
tff(fact_454_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_asym
tff(fact_455_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_asym
tff(fact_456_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_asym
tff(fact_457_linorder__neq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_458_linorder__neq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_459_linorder__neq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_460_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_461_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_462_order__less__asym_H,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2)) ) ).

% order_less_asym'
tff(fact_463_order__less__asym_H,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2)) ) ).

% order_less_asym'
tff(fact_464_order__less__asym_H,axiom,
    ! [A2: num,B3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),A2)) ) ).

% order_less_asym'
tff(fact_465_order__less__asym_H,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2)) ) ).

% order_less_asym'
tff(fact_466_order__less__asym_H,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2)) ) ).

% order_less_asym'
tff(fact_467_order__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z)) ) ) ).

% order_less_trans
tff(fact_468_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).

% order_less_trans
tff(fact_469_order__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_less_trans
tff(fact_470_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).

% order_less_trans
tff(fact_471_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).

% order_less_trans
tff(fact_472_ord__eq__less__subst,axiom,
    ! [A2: real,F: fun_real_real,B3: real,C: real] :
      ( ( A2 = aa_real_real(F,B3) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),C))
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X5)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_473_ord__eq__less__subst,axiom,
    ! [A2: rat,F: fun_real_rat,B3: real,C: real] :
      ( ( A2 = aa_real_rat(F,B3) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),C))
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X5)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_real_rat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_474_ord__eq__less__subst,axiom,
    ! [A2: num,F: fun_real_num,B3: real,C: real] :
      ( ( A2 = aa_real_num(F,B3) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),C))
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X5)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_real_num(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_475_ord__eq__less__subst,axiom,
    ! [A2: nat,F: fun_real_nat,B3: real,C: real] :
      ( ( A2 = aa_real_nat(F,B3) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),C))
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X5)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_real_nat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_476_ord__eq__less__subst,axiom,
    ! [A2: int,F: fun_real_int,B3: real,C: real] :
      ( ( A2 = aa_real_int(F,B3) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),C))
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X5)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_real_int(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_477_ord__eq__less__subst,axiom,
    ! [A2: real,F: fun_rat_real,B3: rat,C: rat] :
      ( ( A2 = aa_rat_real(F,B3) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X5)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_rat_real(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_478_ord__eq__less__subst,axiom,
    ! [A2: rat,F: fun_rat_rat,B3: rat,C: rat] :
      ( ( A2 = aa_rat_rat(F,B3) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_479_ord__eq__less__subst,axiom,
    ! [A2: num,F: fun_rat_num,B3: rat,C: rat] :
      ( ( A2 = aa_rat_num(F,B3) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_rat_num(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_480_ord__eq__less__subst,axiom,
    ! [A2: nat,F: fun_rat_nat,B3: rat,C: rat] :
      ( ( A2 = aa_rat_nat(F,B3) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_rat_nat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_481_ord__eq__less__subst,axiom,
    ! [A2: int,F: fun_rat_int,B3: rat,C: rat] :
      ( ( A2 = aa_rat_int(F,B3) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X5)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_rat_int(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_482_ord__less__eq__subst,axiom,
    ! [A2: real,B3: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( ( aa_real_real(F,B3) = C )
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X5)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_483_ord__less__eq__subst,axiom,
    ! [A2: real,B3: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( ( aa_real_rat(F,B3) = C )
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X5)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_484_ord__less__eq__subst,axiom,
    ! [A2: real,B3: real,F: fun_real_num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( ( aa_real_num(F,B3) = C )
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X5)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_485_ord__less__eq__subst,axiom,
    ! [A2: real,B3: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( ( aa_real_nat(F,B3) = C )
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X5)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_486_ord__less__eq__subst,axiom,
    ! [A2: real,B3: real,F: fun_real_int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( ( aa_real_int(F,B3) = C )
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X5)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_487_ord__less__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( ( aa_rat_real(F,B3) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X5)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_488_ord__less__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( ( aa_rat_rat(F,B3) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_489_ord__less__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( ( aa_rat_num(F,B3) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_490_ord__less__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( ( aa_rat_nat(F,B3) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_491_ord__less__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( ( aa_rat_int(F,B3) = C )
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X5)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_492_order__less__irrefl,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X)) ).

% order_less_irrefl
tff(fact_493_order__less__irrefl,axiom,
    ! [X: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),X)) ).

% order_less_irrefl
tff(fact_494_order__less__irrefl,axiom,
    ! [X: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),X)) ).

% order_less_irrefl
tff(fact_495_order__less__irrefl,axiom,
    ! [X: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X)) ).

% order_less_irrefl
tff(fact_496_order__less__irrefl,axiom,
    ! [X: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X)) ).

% order_less_irrefl
tff(fact_497_order__less__subst1,axiom,
    ! [A2: real,F: fun_real_real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(F,B3)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),C))
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X5)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_498_order__less__subst1,axiom,
    ! [A2: real,F: fun_rat_real,B3: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_rat_real(F,B3)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X5)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_rat_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_499_order__less__subst1,axiom,
    ! [A2: real,F: fun_num_real,B3: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_num_real(F,B3)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X5)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_num_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_500_order__less__subst1,axiom,
    ! [A2: real,F: fun_nat_real,B3: nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_nat_real(F,B3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),C))
       => ( ! [X5: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X5)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_nat_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_501_order__less__subst1,axiom,
    ! [A2: real,F: fun_int_real,B3: int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_int_real(F,B3)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),C))
       => ( ! [X5: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X5)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_int_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_502_order__less__subst1,axiom,
    ! [A2: rat,F: fun_real_rat,B3: real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_real_rat(F,B3)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),C))
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X5)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_real_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_503_order__less__subst1,axiom,
    ! [A2: rat,F: fun_rat_rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(F,B3)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_504_order__less__subst1,axiom,
    ! [A2: rat,F: fun_num_rat,B3: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_num_rat(F,B3)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_num_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_505_order__less__subst1,axiom,
    ! [A2: rat,F: fun_nat_rat,B3: nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_nat_rat(F,B3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),C))
       => ( ! [X5: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X5)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_nat_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_506_order__less__subst1,axiom,
    ! [A2: rat,F: fun_int_rat,B3: int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_int_rat(F,B3)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),C))
       => ( ! [X5: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X5)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_int_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_507_order__less__subst2,axiom,
    ! [A2: real,B3: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B3)),C))
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X5)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_508_order__less__subst2,axiom,
    ! [A2: real,B3: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,B3)),C))
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X5)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_509_order__less__subst2,axiom,
    ! [A2: real,B3: real,F: fun_real_num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,B3)),C))
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X5)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_510_order__less__subst2,axiom,
    ! [A2: real,B3: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,B3)),C))
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X5)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_511_order__less__subst2,axiom,
    ! [A2: real,B3: real,F: fun_real_int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,B3)),C))
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X5)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_512_order__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,B3)),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X5)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_513_order__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,B3)),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_514_order__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B3)),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_515_order__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B3)),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_516_order__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B3)),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X5)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_517_order__less__not__sym,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_not_sym
tff(fact_518_order__less__not__sym,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% order_less_not_sym
tff(fact_519_order__less__not__sym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_not_sym
tff(fact_520_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_not_sym
tff(fact_521_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_not_sym
tff(fact_522_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_523_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_524_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P: bool] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_525_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_526_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_527_linorder__less__linear,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      | ( X = Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_less_linear
tff(fact_528_linorder__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
      | ( X = Y )
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% linorder_less_linear
tff(fact_529_linorder__less__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      | ( X = Y )
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_less_linear
tff(fact_530_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      | ( X = Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_less_linear
tff(fact_531_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      | ( X = Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_less_linear
tff(fact_532_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_533_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_534_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_535_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_536_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_537_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_538_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_539_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_540_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_541_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_542_order__less__imp__not__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_543_order__less__imp__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_544_order__less__imp__not__less,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_545_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_546_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_547_ex__in__conv,axiom,
    ! [A3: set_set_nat] :
      ( ? [X3: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(X3),A3))
    <=> ( A3 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
tff(fact_548_ex__in__conv,axiom,
    ! [A3: set_real] :
      ( ? [X3: real] : pp(aa_set_real_bool(member_real(X3),A3))
    <=> ( A3 != bot_bot_set_real ) ) ).

% ex_in_conv
tff(fact_549_ex__in__conv,axiom,
    ! [A3: set_bool] :
      ( ? [X3: bool] : pp(aa_set_bool_bool(member_bool(X3),A3))
    <=> ( A3 != bot_bot_set_bool ) ) ).

% ex_in_conv
tff(fact_550_ex__in__conv,axiom,
    ! [A3: set_nat] :
      ( ? [X3: nat] : pp(aa_set_nat_bool(member_nat(X3),A3))
    <=> ( A3 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_551_ex__in__conv,axiom,
    ! [A3: set_int] :
      ( ? [X3: int] : pp(aa_set_int_bool(member_int(X3),A3))
    <=> ( A3 != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_552_equals0I,axiom,
    ! [A3: set_set_nat] :
      ( ! [Y3: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(Y3),A3))
     => ( A3 = bot_bot_set_set_nat ) ) ).

% equals0I
tff(fact_553_equals0I,axiom,
    ! [A3: set_real] :
      ( ! [Y3: real] : ~ pp(aa_set_real_bool(member_real(Y3),A3))
     => ( A3 = bot_bot_set_real ) ) ).

% equals0I
tff(fact_554_equals0I,axiom,
    ! [A3: set_bool] :
      ( ! [Y3: bool] : ~ pp(aa_set_bool_bool(member_bool(Y3),A3))
     => ( A3 = bot_bot_set_bool ) ) ).

% equals0I
tff(fact_555_equals0I,axiom,
    ! [A3: set_nat] :
      ( ! [Y3: nat] : ~ pp(aa_set_nat_bool(member_nat(Y3),A3))
     => ( A3 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_556_equals0I,axiom,
    ! [A3: set_int] :
      ( ! [Y3: int] : ~ pp(aa_set_int_bool(member_int(Y3),A3))
     => ( A3 = bot_bot_set_int ) ) ).

% equals0I
tff(fact_557_equals0D,axiom,
    ! [A3: set_set_nat,A2: set_nat] :
      ( ( A3 = bot_bot_set_set_nat )
     => ~ pp(aa_set_set_nat_bool(member_set_nat(A2),A3)) ) ).

% equals0D
tff(fact_558_equals0D,axiom,
    ! [A3: set_real,A2: real] :
      ( ( A3 = bot_bot_set_real )
     => ~ pp(aa_set_real_bool(member_real(A2),A3)) ) ).

% equals0D
tff(fact_559_equals0D,axiom,
    ! [A3: set_bool,A2: bool] :
      ( ( A3 = bot_bot_set_bool )
     => ~ pp(aa_set_bool_bool(member_bool(A2),A3)) ) ).

% equals0D
tff(fact_560_equals0D,axiom,
    ! [A3: set_nat,A2: nat] :
      ( ( A3 = bot_bot_set_nat )
     => ~ pp(aa_set_nat_bool(member_nat(A2),A3)) ) ).

% equals0D
tff(fact_561_equals0D,axiom,
    ! [A3: set_int,A2: int] :
      ( ( A3 = bot_bot_set_int )
     => ~ pp(aa_set_int_bool(member_int(A2),A3)) ) ).

% equals0D
tff(fact_562_emptyE,axiom,
    ! [A2: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(A2),bot_bot_set_set_nat)) ).

% emptyE
tff(fact_563_emptyE,axiom,
    ! [A2: real] : ~ pp(aa_set_real_bool(member_real(A2),bot_bot_set_real)) ).

% emptyE
tff(fact_564_emptyE,axiom,
    ! [A2: bool] : ~ pp(aa_set_bool_bool(member_bool(A2),bot_bot_set_bool)) ).

% emptyE
tff(fact_565_emptyE,axiom,
    ! [A2: nat] : ~ pp(aa_set_nat_bool(member_nat(A2),bot_bot_set_nat)) ).

% emptyE
tff(fact_566_emptyE,axiom,
    ! [A2: int] : ~ pp(aa_set_int_bool(member_int(A2),bot_bot_set_int)) ).

% emptyE
tff(fact_567_not__psubset__empty,axiom,
    ! [A3: set_real] : ~ pp(aa_set_real_bool(ord_less_set_real(A3),bot_bot_set_real)) ).

% not_psubset_empty
tff(fact_568_not__psubset__empty,axiom,
    ! [A3: set_bool] : ~ pp(aa_set_bool_bool(ord_less_set_bool(A3),bot_bot_set_bool)) ).

% not_psubset_empty
tff(fact_569_not__psubset__empty,axiom,
    ! [A3: set_nat] : ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A3),bot_bot_set_nat)) ).

% not_psubset_empty
tff(fact_570_not__psubset__empty,axiom,
    ! [A3: set_int] : ~ pp(aa_set_int_bool(ord_less_set_int(A3),bot_bot_set_int)) ).

% not_psubset_empty
tff(fact_571_finite__psubset__induct,axiom,
    ! [A3: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ! [A4: set_nat] :
            ( pp(aa_set_nat_bool(finite_finite_nat,A4))
           => ( ! [B4: set_nat] :
                  ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B4),A4))
                 => pp(aa_set_nat_bool(P,B4)) )
             => pp(aa_set_nat_bool(P,A4)) ) )
       => pp(aa_set_nat_bool(P,A3)) ) ) ).

% finite_psubset_induct
tff(fact_572_finite__psubset__induct,axiom,
    ! [A3: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ! [A4: set_int] :
            ( pp(aa_set_int_bool(finite_finite_int,A4))
           => ( ! [B4: set_int] :
                  ( pp(aa_set_int_bool(ord_less_set_int(B4),A4))
                 => pp(aa_set_int_bool(P,B4)) )
             => pp(aa_set_int_bool(P,A4)) ) )
       => pp(aa_set_int_bool(P,A3)) ) ) ).

% finite_psubset_induct
tff(fact_573_finite__psubset__induct,axiom,
    ! [A3: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ! [A4: set_complex] :
            ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
           => ( ! [B4: set_complex] :
                  ( pp(aa_set_complex_bool(ord_less_set_complex(B4),A4))
                 => pp(aa_set_complex_bool(P,B4)) )
             => pp(aa_set_complex_bool(P,A4)) ) )
       => pp(aa_set_complex_bool(P,A3)) ) ) ).

% finite_psubset_induct
tff(fact_574_finite__psubset__induct,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( ! [A4: set_Pr1261947904930325089at_nat] :
            ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
           => ( ! [B4: set_Pr1261947904930325089at_nat] :
                  ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(B4),A4))
                 => pp(aa_set3135850010578026034t_bool(P,B4)) )
             => pp(aa_set3135850010578026034t_bool(P,A4)) ) )
       => pp(aa_set3135850010578026034t_bool(P,A3)) ) ) ).

% finite_psubset_induct
tff(fact_575_finite__psubset__induct,axiom,
    ! [A3: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ! [A4: set_Extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A4))
           => ( ! [B4: set_Extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(B4),A4))
                 => pp(aa_set3347977610044113549t_bool(P,B4)) )
             => pp(aa_set3347977610044113549t_bool(P,A4)) ) )
       => pp(aa_set3347977610044113549t_bool(P,A3)) ) ) ).

% finite_psubset_induct
tff(fact_576_leD,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% leD
tff(fact_577_leD,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
     => ~ pp(aa_set_int_bool(ord_less_set_int(X),Y)) ) ).

% leD
tff(fact_578_leD,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% leD
tff(fact_579_leD,axiom,
    ! [Y: num,X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y)) ) ).

% leD
tff(fact_580_leD,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% leD
tff(fact_581_leD,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).

% leD
tff(fact_582_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ).

% leI
tff(fact_583_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% leI
tff(fact_584_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% leI
tff(fact_585_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% leI
tff(fact_586_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% leI
tff(fact_587_nless__le,axiom,
    ! [A2: real,B3: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
        | ( A2 = B3 ) ) ) ).

% nless_le
tff(fact_588_nless__le,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ~ pp(aa_set_int_bool(ord_less_set_int(A2),B3))
    <=> ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
        | ( A2 = B3 ) ) ) ).

% nless_le
tff(fact_589_nless__le,axiom,
    ! [A2: rat,B3: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
    <=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
        | ( A2 = B3 ) ) ) ).

% nless_le
tff(fact_590_nless__le,axiom,
    ! [A2: num,B3: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B3))
    <=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
        | ( A2 = B3 ) ) ) ).

% nless_le
tff(fact_591_nless__le,axiom,
    ! [A2: nat,B3: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
        | ( A2 = B3 ) ) ) ).

% nless_le
tff(fact_592_nless__le,axiom,
    ! [A2: int,B3: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
        | ( A2 = B3 ) ) ) ).

% nless_le
tff(fact_593_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_594_antisym__conv1,axiom,
    ! [X: set_int,Y: set_int] :
      ( ~ pp(aa_set_int_bool(ord_less_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_595_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_596_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_597_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_598_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_599_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_600_antisym__conv2,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( ~ pp(aa_set_int_bool(ord_less_set_int(X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_601_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_602_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_603_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_604_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_605_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X5)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ).

% dense_ge
tff(fact_606_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X5)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ).

% dense_ge
tff(fact_607_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),Z)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ).

% dense_le
tff(fact_608_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Z)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ).

% dense_le
tff(fact_609_less__le__not__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ).

% less_le_not_le
tff(fact_610_less__le__not__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(Y),X)) ) ) ).

% less_le_not_le
tff(fact_611_less__le__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ) ).

% less_le_not_le
tff(fact_612_less__le__not__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ) ).

% less_le_not_le
tff(fact_613_less__le__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ) ).

% less_le_not_le
tff(fact_614_less__le__not__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ) ).

% less_le_not_le
tff(fact_615_not__le__imp__less,axiom,
    ! [Y: real,X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% not_le_imp_less
tff(fact_616_not__le__imp__less,axiom,
    ! [Y: rat,X: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% not_le_imp_less
tff(fact_617_not__le__imp__less,axiom,
    ! [Y: num,X: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y)) ) ).

% not_le_imp_less
tff(fact_618_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% not_le_imp_less
tff(fact_619_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).

% not_le_imp_less
tff(fact_620_order_Oorder__iff__strict,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
        | ( A2 = B3 ) ) ) ).

% order.order_iff_strict
tff(fact_621_order_Oorder__iff__strict,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(A2),B3))
        | ( A2 = B3 ) ) ) ).

% order.order_iff_strict
tff(fact_622_order_Oorder__iff__strict,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
        | ( A2 = B3 ) ) ) ).

% order.order_iff_strict
tff(fact_623_order_Oorder__iff__strict,axiom,
    ! [A2: num,B3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B3))
        | ( A2 = B3 ) ) ) ).

% order.order_iff_strict
tff(fact_624_order_Oorder__iff__strict,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
        | ( A2 = B3 ) ) ) ).

% order.order_iff_strict
tff(fact_625_order_Oorder__iff__strict,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
        | ( A2 = B3 ) ) ) ).

% order.order_iff_strict
tff(fact_626_order_Ostrict__iff__order,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
        & ( A2 != B3 ) ) ) ).

% order.strict_iff_order
tff(fact_627_order_Ostrict__iff__order,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B3))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
        & ( A2 != B3 ) ) ) ).

% order.strict_iff_order
tff(fact_628_order_Ostrict__iff__order,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
        & ( A2 != B3 ) ) ) ).

% order.strict_iff_order
tff(fact_629_order_Ostrict__iff__order,axiom,
    ! [A2: num,B3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B3))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
        & ( A2 != B3 ) ) ) ).

% order.strict_iff_order
tff(fact_630_order_Ostrict__iff__order,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
        & ( A2 != B3 ) ) ) ).

% order.strict_iff_order
tff(fact_631_order_Ostrict__iff__order,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
        & ( A2 != B3 ) ) ) ).

% order.strict_iff_order
tff(fact_632_order_Ostrict__trans1,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C)) ) ) ).

% order.strict_trans1
tff(fact_633_order_Ostrict__trans1,axiom,
    ! [A2: set_int,B3: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( pp(aa_set_int_bool(ord_less_set_int(B3),C))
       => pp(aa_set_int_bool(ord_less_set_int(A2),C)) ) ) ).

% order.strict_trans1
tff(fact_634_order_Ostrict__trans1,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),C)) ) ) ).

% order.strict_trans1
tff(fact_635_order_Ostrict__trans1,axiom,
    ! [A2: num,B3: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),C)) ) ) ).

% order.strict_trans1
tff(fact_636_order_Ostrict__trans1,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C)) ) ) ).

% order.strict_trans1
tff(fact_637_order_Ostrict__trans1,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C)) ) ) ).

% order.strict_trans1
tff(fact_638_order_Ostrict__trans2,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C)) ) ) ).

% order.strict_trans2
tff(fact_639_order_Ostrict__trans2,axiom,
    ! [A2: set_int,B3: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),C))
       => pp(aa_set_int_bool(ord_less_set_int(A2),C)) ) ) ).

% order.strict_trans2
tff(fact_640_order_Ostrict__trans2,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),C)) ) ) ).

% order.strict_trans2
tff(fact_641_order_Ostrict__trans2,axiom,
    ! [A2: num,B3: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),C)) ) ) ).

% order.strict_trans2
tff(fact_642_order_Ostrict__trans2,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C)) ) ) ).

% order.strict_trans2
tff(fact_643_order_Ostrict__trans2,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C)) ) ) ).

% order.strict_trans2
tff(fact_644_order_Ostrict__iff__not,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2)) ) ) ).

% order.strict_iff_not
tff(fact_645_order_Ostrict__iff__not,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B3))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2)) ) ) ).

% order.strict_iff_not
tff(fact_646_order_Ostrict__iff__not,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2)) ) ) ).

% order.strict_iff_not
tff(fact_647_order_Ostrict__iff__not,axiom,
    ! [A2: num,B3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B3))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2)) ) ) ).

% order.strict_iff_not
tff(fact_648_order_Ostrict__iff__not,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2)) ) ) ).

% order.strict_iff_not
tff(fact_649_order_Ostrict__iff__not,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2)) ) ) ).

% order.strict_iff_not
tff(fact_650_dense__ge__bounded,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X))
     => ( ! [W: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),W))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W),X))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),W)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ) ).

% dense_ge_bounded
tff(fact_651_dense__ge__bounded,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X))
     => ( ! [W: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),W))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W),X))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),W)) ) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ) ).

% dense_ge_bounded
tff(fact_652_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( ! [W: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),W))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W),Y))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W),Z)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ) ).

% dense_le_bounded
tff(fact_653_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( ! [W: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),W))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W),Y))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W),Z)) ) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ) ).

% dense_le_bounded
tff(fact_654_dual__order_Oorder__iff__strict,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
        | ( A2 = B3 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_655_dual__order_Oorder__iff__strict,axiom,
    ! [B3: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(B3),A2))
        | ( A2 = B3 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_656_dual__order_Oorder__iff__strict,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
        | ( A2 = B3 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_657_dual__order_Oorder__iff__strict,axiom,
    ! [B3: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),A2))
        | ( A2 = B3 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_658_dual__order_Oorder__iff__strict,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
        | ( A2 = B3 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_659_dual__order_Oorder__iff__strict,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
        | ( A2 = B3 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_660_dual__order_Ostrict__iff__order,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2))
        & ( A2 != B3 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_661_dual__order_Ostrict__iff__order,axiom,
    ! [B3: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B3),A2))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
        & ( A2 != B3 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_662_dual__order_Ostrict__iff__order,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
        & ( A2 != B3 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_663_dual__order_Ostrict__iff__order,axiom,
    ! [B3: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),A2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2))
        & ( A2 != B3 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_664_dual__order_Ostrict__iff__order,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
        & ( A2 != B3 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_665_dual__order_Ostrict__iff__order,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
        & ( A2 != B3 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_666_dual__order_Ostrict__trans1,axiom,
    ! [B3: real,A2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A2)) ) ) ).

% dual_order.strict_trans1
tff(fact_667_dual__order_Ostrict__trans1,axiom,
    ! [B3: set_int,A2: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
     => ( pp(aa_set_int_bool(ord_less_set_int(C),B3))
       => pp(aa_set_int_bool(ord_less_set_int(C),A2)) ) ) ).

% dual_order.strict_trans1
tff(fact_668_dual__order_Ostrict__trans1,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A2)) ) ) ).

% dual_order.strict_trans1
tff(fact_669_dual__order_Ostrict__trans1,axiom,
    ! [B3: num,A2: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A2)) ) ) ).

% dual_order.strict_trans1
tff(fact_670_dual__order_Ostrict__trans1,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A2)) ) ) ).

% dual_order.strict_trans1
tff(fact_671_dual__order_Ostrict__trans1,axiom,
    ! [B3: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A2)) ) ) ).

% dual_order.strict_trans1
tff(fact_672_dual__order_Ostrict__trans2,axiom,
    ! [B3: real,A2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A2)) ) ) ).

% dual_order.strict_trans2
tff(fact_673_dual__order_Ostrict__trans2,axiom,
    ! [B3: set_int,A2: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B3),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(C),B3))
       => pp(aa_set_int_bool(ord_less_set_int(C),A2)) ) ) ).

% dual_order.strict_trans2
tff(fact_674_dual__order_Ostrict__trans2,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A2)) ) ) ).

% dual_order.strict_trans2
tff(fact_675_dual__order_Ostrict__trans2,axiom,
    ! [B3: num,A2: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),A2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A2)) ) ) ).

% dual_order.strict_trans2
tff(fact_676_dual__order_Ostrict__trans2,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A2)) ) ) ).

% dual_order.strict_trans2
tff(fact_677_dual__order_Ostrict__trans2,axiom,
    ! [B3: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A2)) ) ) ).

% dual_order.strict_trans2
tff(fact_678_dual__order_Ostrict__iff__not,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ) ).

% dual_order.strict_iff_not
tff(fact_679_dual__order_Ostrict__iff__not,axiom,
    ! [B3: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B3),A2))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3)) ) ) ).

% dual_order.strict_iff_not
tff(fact_680_dual__order_Ostrict__iff__not,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ) ).

% dual_order.strict_iff_not
tff(fact_681_dual__order_Ostrict__iff__not,axiom,
    ! [B3: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),A2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3)) ) ) ).

% dual_order.strict_iff_not
tff(fact_682_dual__order_Ostrict__iff__not,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3)) ) ) ).

% dual_order.strict_iff_not
tff(fact_683_dual__order_Ostrict__iff__not,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ) ).

% dual_order.strict_iff_not
tff(fact_684_order_Ostrict__implies__order,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ).

% order.strict_implies_order
tff(fact_685_order_Ostrict__implies__order,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B3))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3)) ) ).

% order.strict_implies_order
tff(fact_686_order_Ostrict__implies__order,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ).

% order.strict_implies_order
tff(fact_687_order_Ostrict__implies__order,axiom,
    ! [A2: num,B3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3)) ) ).

% order.strict_implies_order
tff(fact_688_order_Ostrict__implies__order,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3)) ) ).

% order.strict_implies_order
tff(fact_689_order_Ostrict__implies__order,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ).

% order.strict_implies_order
tff(fact_690_dual__order_Ostrict__implies__order,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2)) ) ).

% dual_order.strict_implies_order
tff(fact_691_dual__order_Ostrict__implies__order,axiom,
    ! [B3: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B3),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2)) ) ).

% dual_order.strict_implies_order
tff(fact_692_dual__order_Ostrict__implies__order,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2)) ) ).

% dual_order.strict_implies_order
tff(fact_693_dual__order_Ostrict__implies__order,axiom,
    ! [B3: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),A2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2)) ) ).

% dual_order.strict_implies_order
tff(fact_694_dual__order_Ostrict__implies__order,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2)) ) ).

% dual_order.strict_implies_order
tff(fact_695_dual__order_Ostrict__implies__order,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2)) ) ).

% dual_order.strict_implies_order
tff(fact_696_order__le__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_697_order__le__less,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_698_order__le__less,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_699_order__le__less,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_700_order__le__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_701_order__le__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_702_order__less__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_703_order__less__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_704_order__less__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_705_order__less__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_706_order__less__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_707_order__less__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_708_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_not_le
tff(fact_709_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% linorder_not_le
tff(fact_710_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_not_le
tff(fact_711_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_not_le
tff(fact_712_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_not_le
tff(fact_713_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ).

% linorder_not_less
tff(fact_714_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% linorder_not_less
tff(fact_715_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% linorder_not_less
tff(fact_716_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_not_less
tff(fact_717_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_not_less
tff(fact_718_order__less__imp__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% order_less_imp_le
tff(fact_719_order__less__imp__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
     => pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).

% order_less_imp_le
tff(fact_720_order__less__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% order_less_imp_le
tff(fact_721_order__less__imp__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% order_less_imp_le
tff(fact_722_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% order_less_imp_le
tff(fact_723_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% order_less_imp_le
tff(fact_724_order__le__neq__trans,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( ( A2 != B3 )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) ) ) ).

% order_le_neq_trans
tff(fact_725_order__le__neq__trans,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( ( A2 != B3 )
       => pp(aa_set_int_bool(ord_less_set_int(A2),B3)) ) ) ).

% order_le_neq_trans
tff(fact_726_order__le__neq__trans,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( ( A2 != B3 )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) ) ) ).

% order_le_neq_trans
tff(fact_727_order__le__neq__trans,axiom,
    ! [A2: num,B3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( ( A2 != B3 )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B3)) ) ) ).

% order_le_neq_trans
tff(fact_728_order__le__neq__trans,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( ( A2 != B3 )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3)) ) ) ).

% order_le_neq_trans
tff(fact_729_order__le__neq__trans,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( ( A2 != B3 )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3)) ) ) ).

% order_le_neq_trans
tff(fact_730_order__neq__le__trans,axiom,
    ! [A2: real,B3: real] :
      ( ( A2 != B3 )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) ) ) ).

% order_neq_le_trans
tff(fact_731_order__neq__le__trans,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 != B3 )
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
       => pp(aa_set_int_bool(ord_less_set_int(A2),B3)) ) ) ).

% order_neq_le_trans
tff(fact_732_order__neq__le__trans,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 != B3 )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) ) ) ).

% order_neq_le_trans
tff(fact_733_order__neq__le__trans,axiom,
    ! [A2: num,B3: num] :
      ( ( A2 != B3 )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B3)) ) ) ).

% order_neq_le_trans
tff(fact_734_order__neq__le__trans,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 != B3 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3)) ) ) ).

% order_neq_le_trans
tff(fact_735_order__neq__le__trans,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 != B3 )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3)) ) ) ).

% order_neq_le_trans
tff(fact_736_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_737_order__le__less__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_set_int(Y),Z))
       => pp(aa_set_int_bool(ord_less_set_int(X),Z)) ) ) ).

% order_le_less_trans
tff(fact_738_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_739_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_740_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_741_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_742_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_743_order__less__le__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),Z))
       => pp(aa_set_int_bool(ord_less_set_int(X),Z)) ) ) ).

% order_less_le_trans
tff(fact_744_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_745_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_746_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_747_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_748_order__le__less__subst1,axiom,
    ! [A2: real,F: fun_real_real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(F,B3)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),C))
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X5)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_749_order__le__less__subst1,axiom,
    ! [A2: real,F: fun_rat_real,B3: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_rat_real(F,B3)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X5)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_rat_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_750_order__le__less__subst1,axiom,
    ! [A2: real,F: fun_num_real,B3: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_num_real(F,B3)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X5)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_num_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_751_order__le__less__subst1,axiom,
    ! [A2: real,F: fun_nat_real,B3: nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_nat_real(F,B3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),C))
       => ( ! [X5: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X5)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_nat_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_752_order__le__less__subst1,axiom,
    ! [A2: real,F: fun_int_real,B3: int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_int_real(F,B3)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),C))
       => ( ! [X5: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X5)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_int_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_753_order__le__less__subst1,axiom,
    ! [A2: rat,F: fun_real_rat,B3: real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_real_rat(F,B3)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),C))
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X5)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_real_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_754_order__le__less__subst1,axiom,
    ! [A2: rat,F: fun_rat_rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(F,B3)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_755_order__le__less__subst1,axiom,
    ! [A2: rat,F: fun_num_rat,B3: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_num_rat(F,B3)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_num_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_756_order__le__less__subst1,axiom,
    ! [A2: rat,F: fun_nat_rat,B3: nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_nat_rat(F,B3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),C))
       => ( ! [X5: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X5)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_nat_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_757_order__le__less__subst1,axiom,
    ! [A2: rat,F: fun_int_rat,B3: int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_int_rat(F,B3)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),C))
       => ( ! [X5: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X5)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_int_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_758_order__le__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,B3)),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X5)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_759_order__le__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,B3)),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_760_order__le__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B3)),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_761_order__le__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B3)),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_762_order__le__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B3)),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X5)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_763_order__le__less__subst2,axiom,
    ! [A2: num,B3: num,F: fun_num_real,C: real] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,B3)),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X5)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_764_order__le__less__subst2,axiom,
    ! [A2: num,B3: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,B3)),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_765_order__le__less__subst2,axiom,
    ! [A2: num,B3: num,F: fun_num_num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,B3)),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X5)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_766_order__le__less__subst2,axiom,
    ! [A2: num,B3: num,F: fun_num_nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,B3)),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X5)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_767_order__le__less__subst2,axiom,
    ! [A2: num,B3: num,F: fun_num_int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,B3)),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X5)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_768_order__less__le__subst1,axiom,
    ! [A2: real,F: fun_rat_real,B3: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_rat_real(F,B3)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X5)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_rat_real(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_769_order__less__le__subst1,axiom,
    ! [A2: rat,F: fun_rat_rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(F,B3)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_770_order__less__le__subst1,axiom,
    ! [A2: num,F: fun_rat_num,B3: rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_rat_num(F,B3)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X5)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_rat_num(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_771_order__less__le__subst1,axiom,
    ! [A2: nat,F: fun_rat_nat,B3: rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_rat_nat(F,B3)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X5)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_rat_nat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_772_order__less__le__subst1,axiom,
    ! [A2: int,F: fun_rat_int,B3: rat,C: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_rat_int(F,B3)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X5)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_rat_int(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_773_order__less__le__subst1,axiom,
    ! [A2: real,F: fun_num_real,B3: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_num_real(F,B3)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X5)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_num_real(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_774_order__less__le__subst1,axiom,
    ! [A2: rat,F: fun_num_rat,B3: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_num_rat(F,B3)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_num_rat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_775_order__less__le__subst1,axiom,
    ! [A2: num,F: fun_num_num,B3: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_num_num(F,B3)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X5)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_num_num(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_776_order__less__le__subst1,axiom,
    ! [A2: nat,F: fun_num_nat,B3: num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_num_nat(F,B3)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X5)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_num_nat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_777_order__less__le__subst1,axiom,
    ! [A2: int,F: fun_num_int,B3: num,C: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_num_int(F,B3)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X5)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_num_int(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_778_order__less__le__subst2,axiom,
    ! [A2: real,B3: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,B3)),C))
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X5)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_779_order__less__le__subst2,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,B3)),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X5)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_780_order__less__le__subst2,axiom,
    ! [A2: num,B3: num,F: fun_num_real,C: real] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,B3)),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X5)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_781_order__less__le__subst2,axiom,
    ! [A2: nat,B3: nat,F: fun_nat_real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,B3)),C))
       => ( ! [X5: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X5)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_782_order__less__le__subst2,axiom,
    ! [A2: int,B3: int,F: fun_int_real,C: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,B3)),C))
       => ( ! [X5: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X5)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_783_order__less__le__subst2,axiom,
    ! [A2: real,B3: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,B3)),C))
       => ( ! [X5: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X5)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_784_order__less__le__subst2,axiom,
    ! [A2: rat,B3: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,B3)),C))
       => ( ! [X5: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X5)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_785_order__less__le__subst2,axiom,
    ! [A2: num,B3: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,B3)),C))
       => ( ! [X5: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X5)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_786_order__less__le__subst2,axiom,
    ! [A2: nat,B3: nat,F: fun_nat_rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,B3)),C))
       => ( ! [X5: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X5)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_787_order__less__le__subst2,axiom,
    ! [A2: int,B3: int,F: fun_int_rat,C: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,B3)),C))
       => ( ! [X5: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X5)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_788_linorder__le__less__linear,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_789_linorder__le__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_790_linorder__le__less__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_791_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_792_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_793_order__le__imp__less__or__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_794_order__le__imp__less__or__eq,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_795_order__le__imp__less__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_796_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_797_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_798_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_799_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),bot_bot_set_real))
     => ( A2 = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_800_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),bot_bot_set_bool))
     => ( A2 = bot_bot_set_bool ) ) ).

% bot.extremum_uniqueI
tff(fact_801_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),bot_bot_set_nat))
     => ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_802_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),bot_bot_set_int))
     => ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_803_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),bot_bot_nat))
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_804_bot_Oextremum__unique,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),bot_bot_set_real))
    <=> ( A2 = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_805_bot_Oextremum__unique,axiom,
    ! [A2: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),bot_bot_set_bool))
    <=> ( A2 = bot_bot_set_bool ) ) ).

% bot.extremum_unique
tff(fact_806_bot_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),bot_bot_set_nat))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_807_bot_Oextremum__unique,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),bot_bot_set_int))
    <=> ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_808_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),bot_bot_nat))
    <=> ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_809_bot_Oextremum,axiom,
    ! [A2: set_real] : pp(aa_set_real_bool(ord_less_eq_set_real(bot_bot_set_real),A2)) ).

% bot.extremum
tff(fact_810_bot_Oextremum,axiom,
    ! [A2: set_bool] : pp(aa_set_bool_bool(ord_less_eq_set_bool(bot_bot_set_bool),A2)) ).

% bot.extremum
tff(fact_811_bot_Oextremum,axiom,
    ! [A2: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,bot_bot_set_nat),A2)) ).

% bot.extremum
tff(fact_812_bot_Oextremum,axiom,
    ! [A2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(bot_bot_set_int),A2)) ).

% bot.extremum
tff(fact_813_bot_Oextremum,axiom,
    ! [A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,bot_bot_nat),A2)) ).

% bot.extremum
tff(fact_814_bot_Oextremum__strict,axiom,
    ! [A2: set_real] : ~ pp(aa_set_real_bool(ord_less_set_real(A2),bot_bot_set_real)) ).

% bot.extremum_strict
tff(fact_815_bot_Oextremum__strict,axiom,
    ! [A2: set_bool] : ~ pp(aa_set_bool_bool(ord_less_set_bool(A2),bot_bot_set_bool)) ).

% bot.extremum_strict
tff(fact_816_bot_Oextremum__strict,axiom,
    ! [A2: set_nat] : ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),bot_bot_set_nat)) ).

% bot.extremum_strict
tff(fact_817_bot_Oextremum__strict,axiom,
    ! [A2: set_int] : ~ pp(aa_set_int_bool(ord_less_set_int(A2),bot_bot_set_int)) ).

% bot.extremum_strict
tff(fact_818_bot_Oextremum__strict,axiom,
    ! [A2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),bot_bot_nat)) ).

% bot.extremum_strict
tff(fact_819_bot_Onot__eq__extremum,axiom,
    ! [A2: set_real] :
      ( ( A2 != bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_set_real(bot_bot_set_real),A2)) ) ).

% bot.not_eq_extremum
tff(fact_820_bot_Onot__eq__extremum,axiom,
    ! [A2: set_bool] :
      ( ( A2 != bot_bot_set_bool )
    <=> pp(aa_set_bool_bool(ord_less_set_bool(bot_bot_set_bool),A2)) ) ).

% bot.not_eq_extremum
tff(fact_821_bot_Onot__eq__extremum,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,bot_bot_set_nat),A2)) ) ).

% bot.not_eq_extremum
tff(fact_822_bot_Onot__eq__extremum,axiom,
    ! [A2: set_int] :
      ( ( A2 != bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_set_int(bot_bot_set_int),A2)) ) ).

% bot.not_eq_extremum
tff(fact_823_bot_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != bot_bot_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,bot_bot_nat),A2)) ) ).

% bot.not_eq_extremum
tff(fact_824_finite__has__minimal2,axiom,
    ! [A3: set_real,A2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(member_real(A2),A3))
       => ? [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),A3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),A2))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X5))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_825_finite__has__minimal2,axiom,
    ! [A3: set_bool,A2: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( pp(aa_set_bool_bool(member_bool(A2),A3))
       => ? [X5: bool] :
            ( pp(aa_set_bool_bool(member_bool(X5),A3))
            & pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,X5),A2))
            & ! [Xa: bool] :
                ( pp(aa_set_bool_bool(member_bool(Xa),A3))
               => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,Xa),X5))
                 => ( pp(X5)
                  <=> pp(Xa) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_826_finite__has__minimal2,axiom,
    ! [A3: set_set_nat,A2: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( pp(aa_set_set_nat_bool(member_set_nat(A2),A3))
       => ? [X5: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X5),A3))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X5),A2))
            & ! [Xa: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(Xa),A3))
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Xa),X5))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_827_finite__has__minimal2,axiom,
    ! [A3: set_Extended_enat,A2: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),A3))
       => ? [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A3))
            & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X5),A2))
            & ! [Xa: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),A3))
               => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,Xa),X5))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_828_finite__has__minimal2,axiom,
    ! [A3: set_set_int,A2: set_int] :
      ( finite6197958912794628473et_int(A3)
     => ( member_set_int(A2,A3)
       => ? [X5: set_int] :
            ( member_set_int(X5,A3)
            & pp(aa_set_int_bool(ord_less_eq_set_int(X5),A2))
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A3)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(Xa),X5))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_829_finite__has__minimal2,axiom,
    ! [A3: set_rat,A2: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A3))
     => ( pp(aa_set_rat_bool(member_rat(A2),A3))
       => ? [X5: rat] :
            ( pp(aa_set_rat_bool(member_rat(X5),A3))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),A2))
            & ! [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),A3))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa),X5))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_830_finite__has__minimal2,axiom,
    ! [A3: set_num,A2: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A3))
     => ( pp(aa_set_num_bool(member_num(A2),A3))
       => ? [X5: num] :
            ( pp(aa_set_num_bool(member_num(X5),A3))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),A2))
            & ! [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),A3))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa),X5))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_831_finite__has__minimal2,axiom,
    ! [A3: set_nat,A2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(member_nat(A2),A3))
       => ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),A2))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A3))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X5))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_832_finite__has__minimal2,axiom,
    ! [A3: set_int,A2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(member_int(A2),A3))
       => ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A3))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),A2))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A3))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X5))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_833_finite__has__maximal2,axiom,
    ! [A3: set_real,A2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(member_real(A2),A3))
       => ? [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),A3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X5))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),Xa))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_834_finite__has__maximal2,axiom,
    ! [A3: set_bool,A2: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( pp(aa_set_bool_bool(member_bool(A2),A3))
       => ? [X5: bool] :
            ( pp(aa_set_bool_bool(member_bool(X5),A3))
            & pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,A2),X5))
            & ! [Xa: bool] :
                ( pp(aa_set_bool_bool(member_bool(Xa),A3))
               => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,X5),Xa))
                 => ( pp(X5)
                  <=> pp(Xa) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_835_finite__has__maximal2,axiom,
    ! [A3: set_set_nat,A2: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( pp(aa_set_set_nat_bool(member_set_nat(A2),A3))
       => ? [X5: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X5),A3))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),X5))
            & ! [Xa: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(Xa),A3))
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X5),Xa))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_836_finite__has__maximal2,axiom,
    ! [A3: set_Extended_enat,A2: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),A3))
       => ? [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A3))
            & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,A2),X5))
            & ! [Xa: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),A3))
               => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,X5),Xa))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_837_finite__has__maximal2,axiom,
    ! [A3: set_set_int,A2: set_int] :
      ( finite6197958912794628473et_int(A3)
     => ( member_set_int(A2,A3)
       => ? [X5: set_int] :
            ( member_set_int(X5,A3)
            & pp(aa_set_int_bool(ord_less_eq_set_int(A2),X5))
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A3)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(X5),Xa))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_838_finite__has__maximal2,axiom,
    ! [A3: set_rat,A2: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A3))
     => ( pp(aa_set_rat_bool(member_rat(A2),A3))
       => ? [X5: rat] :
            ( pp(aa_set_rat_bool(member_rat(X5),A3))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),X5))
            & ! [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),A3))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Xa))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_839_finite__has__maximal2,axiom,
    ! [A3: set_num,A2: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A3))
     => ( pp(aa_set_num_bool(member_num(A2),A3))
       => ? [X5: num] :
            ( pp(aa_set_num_bool(member_num(X5),A3))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),X5))
            & ! [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),A3))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Xa))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_840_finite__has__maximal2,axiom,
    ! [A3: set_nat,A2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(member_nat(A2),A3))
       => ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),X5))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A3))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Xa))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_841_finite__has__maximal2,axiom,
    ! [A3: set_int,A2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(member_int(A2),A3))
       => ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A3))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),X5))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A3))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),Xa))
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_842_infinite__imp__nonempty,axiom,
    ! [S: set_complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( S != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
tff(fact_843_infinite__imp__nonempty,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S))
     => ( S != bot_bo2099793752762293965at_nat ) ) ).

% infinite_imp_nonempty
tff(fact_844_infinite__imp__nonempty,axiom,
    ! [S: set_Extended_enat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( S != bot_bo7653980558646680370d_enat ) ) ).

% infinite_imp_nonempty
tff(fact_845_infinite__imp__nonempty,axiom,
    ! [S: set_real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,S))
     => ( S != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
tff(fact_846_infinite__imp__nonempty,axiom,
    ! [S: set_bool] :
      ( ~ pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( S != bot_bot_set_bool ) ) ).

% infinite_imp_nonempty
tff(fact_847_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
tff(fact_848_infinite__imp__nonempty,axiom,
    ! [S: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S))
     => ( S != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
tff(fact_849_finite_OemptyI,axiom,
    pp(aa_set_complex_bool(finite3207457112153483333omplex,bot_bot_set_complex)) ).

% finite.emptyI
tff(fact_850_finite_OemptyI,axiom,
    pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,bot_bo2099793752762293965at_nat)) ).

% finite.emptyI
tff(fact_851_finite_OemptyI,axiom,
    pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,bot_bo7653980558646680370d_enat)) ).

% finite.emptyI
tff(fact_852_finite_OemptyI,axiom,
    pp(aa_set_real_bool(finite_finite_real,bot_bot_set_real)) ).

% finite.emptyI
tff(fact_853_finite_OemptyI,axiom,
    pp(aa_set_bool_bool(finite_finite_bool,bot_bot_set_bool)) ).

% finite.emptyI
tff(fact_854_finite_OemptyI,axiom,
    pp(aa_set_nat_bool(finite_finite_nat,bot_bot_set_nat)) ).

% finite.emptyI
tff(fact_855_finite_OemptyI,axiom,
    pp(aa_set_int_bool(finite_finite_int,bot_bot_set_int)) ).

% finite.emptyI
tff(fact_856_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),N))
     => ( pp(aa_nat_bool(vEBT_vebt_member(Tree),X))
       => ( pp(aa_nat_bool(vEBT_V5719532721284313246member(Tree),X))
          | pp(aa_nat_bool(vEBT_VEBT_membermima(Tree),X)) ) ) ) ).

% member_valid_both_member_options
tff(fact_857_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_min_max
tff(fact_858_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_vebt_delete(T,X)),Y))
      <=> ( ( X != Y )
          & pp(aa_nat_bool(vEBT_vebt_member(T),Y)) ) ) ) ).

% dele_member_cont_corr
tff(fact_859_finite__nat__set__iff__bounded__le,axiom,
    ! [N4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
    <=> ? [M3: nat] :
        ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),M3)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_860_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
    <=> ! [M3: nat] :
        ? [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N3))
          & pp(aa_set_nat_bool(member_nat(N3),S)) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_861_unbounded__k__infinite,axiom,
    ! [K: nat,S: set_nat] :
      ( ! [M4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),M4))
         => ? [N5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N5))
              & pp(aa_set_nat_bool(member_nat(N5),S)) ) )
     => ~ pp(aa_set_nat_bool(finite_finite_nat,S)) ) ).

% unbounded_k_infinite
tff(fact_862_bounded__nat__set__is__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),N)) )
     => pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).

% bounded_nat_set_is_finite
tff(fact_863_infinite__nat__iff__unbounded,axiom,
    ! [S: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
    <=> ! [M3: nat] :
        ? [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N3))
          & pp(aa_set_nat_bool(member_nat(N3),S)) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_864_finite__nat__set__iff__bounded,axiom,
    ! [N4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
    <=> ? [M3: nat] :
        ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),M3)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_865_arg__min__if__finite_I2_J,axiom,
    ! [S: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( S != bot_bot_set_complex )
       => ~ ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),S))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X4)),aa_complex_real(F,lattic8794016678065449205x_real(F,S)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_866_arg__min__if__finite_I2_J,axiom,
    ! [S: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ( S != bot_bo7653980558646680370d_enat )
       => ~ ? [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(F,lattic1189837152898106425t_real(F,S)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_867_arg__min__if__finite_I2_J,axiom,
    ! [S: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( S != bot_bot_set_real )
       => ~ ? [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),S))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,lattic8440615504127631091l_real(F,S)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_868_arg__min__if__finite_I2_J,axiom,
    ! [S: set_bool,F: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( ( S != bot_bot_set_bool )
       => ~ ? [X4: bool] :
              ( pp(aa_set_bool_bool(member_bool(X4),S))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_bool_real(F,X4)),aa_bool_real(F,aa_set_bool_bool(lattic5237676299781464426l_real(F),S)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_869_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),S))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,aa_set_nat_nat(lattic488527866317076247t_real(F),S)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_870_arg__min__if__finite_I2_J,axiom,
    ! [S: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ( S != bot_bot_set_int )
       => ~ ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),S))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,lattic2675449441010098035t_real(F,S)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_871_arg__min__if__finite_I2_J,axiom,
    ! [S: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( S != bot_bot_set_complex )
       => ~ ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),S))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X4)),aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_872_arg__min__if__finite_I2_J,axiom,
    ! [S: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ( S != bot_bo7653980558646680370d_enat )
       => ~ ? [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Extended_enat_rat(F,X4)),aa_Extended_enat_rat(F,lattic3210252021154270693at_rat(F,S)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_873_arg__min__if__finite_I2_J,axiom,
    ! [S: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( S != bot_bot_set_real )
       => ~ ? [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),S))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,lattic4420706379359479199al_rat(F,S)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_874_arg__min__if__finite_I2_J,axiom,
    ! [S: set_bool,F: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( ( S != bot_bot_set_bool )
       => ~ ? [X4: bool] :
              ( pp(aa_set_bool_bool(member_bool(X4),S))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_bool_rat(F,X4)),aa_bool_rat(F,aa_set_bool_bool(lattic6202137068483785622ol_rat(F),S)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_875_arg__min__least,axiom,
    ! [S: set_complex,Y: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( S != bot_bot_set_complex )
       => ( pp(aa_set_complex_bool(member_complex(Y),S))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S))),aa_complex_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_876_arg__min__least,axiom,
    ! [S: set_Extended_enat,Y: extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ( S != bot_bo7653980558646680370d_enat )
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y),S))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Extended_enat_rat(F,lattic3210252021154270693at_rat(F,S))),aa_Extended_enat_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_877_arg__min__least,axiom,
    ! [S: set_real,Y: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( S != bot_bot_set_real )
       => ( pp(aa_set_real_bool(member_real(Y),S))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,lattic4420706379359479199al_rat(F,S))),aa_real_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_878_arg__min__least,axiom,
    ! [S: set_bool,Y: bool,F: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( ( S != bot_bot_set_bool )
       => ( pp(aa_set_bool_bool(member_bool(Y),S))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_bool_rat(F,aa_set_bool_bool(lattic6202137068483785622ol_rat(F),S))),aa_bool_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_879_arg__min__least,axiom,
    ! [S: set_nat,Y: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ( S != bot_bot_set_nat )
       => ( pp(aa_set_nat_bool(member_nat(Y),S))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S))),aa_nat_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_880_arg__min__least,axiom,
    ! [S: set_int,Y: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ( S != bot_bot_set_int )
       => ( pp(aa_set_int_bool(member_int(Y),S))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,lattic7811156612396918303nt_rat(F,S))),aa_int_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_881_arg__min__least,axiom,
    ! [S: set_complex,Y: complex,F: fun_complex_num] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( S != bot_bot_set_complex )
       => ( pp(aa_set_complex_bool(member_complex(Y),S))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_complex_num(F,lattic1922116423962787043ex_num(F,S))),aa_complex_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_882_arg__min__least,axiom,
    ! [S: set_Extended_enat,Y: extended_enat,F: fun_Ex8368542335153306515at_num] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ( S != bot_bo7653980558646680370d_enat )
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y),S))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_Extended_enat_num(F,lattic402713867396545063at_num(F,S))),aa_Extended_enat_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_883_arg__min__least,axiom,
    ! [S: set_real,Y: real,F: fun_real_num] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( S != bot_bot_set_real )
       => ( pp(aa_set_real_bool(member_real(Y),S))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_real_num(F,lattic1613168225601753569al_num(F,S))),aa_real_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_884_arg__min__least,axiom,
    ! [S: set_bool,Y: bool,F: fun_bool_num] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( ( S != bot_bot_set_bool )
       => ( pp(aa_set_bool_bool(member_bool(Y),S))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_bool_num(F,aa_set_bool_bool(lattic3394598914726059992ol_num(F),S))),aa_bool_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_885_nat__descend__induct,axiom,
    ! [N: nat,P: fun_nat_bool,M: nat] :
      ( ! [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
         => pp(aa_nat_bool(P,K2)) )
     => ( ! [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
           => ( ! [I3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),I3))
                 => pp(aa_nat_bool(P,I3)) )
             => pp(aa_nat_bool(P,K2)) ) )
       => pp(aa_nat_bool(P,M)) ) ) ).

% nat_descend_induct
tff(fact_886_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_delete(T,X)),N)) ) ).

% delete_pres_valid
tff(fact_887_subsetI,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),A3))
         => pp(aa_set_real_bool(member_real(X5),B2)) )
     => pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2)) ) ).

% subsetI
tff(fact_888_subsetI,axiom,
    ! [A3: set_bool,B2: set_bool] :
      ( ! [X5: bool] :
          ( pp(aa_set_bool_bool(member_bool(X5),A3))
         => pp(aa_set_bool_bool(member_bool(X5),B2)) )
     => pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),B2)) ) ).

% subsetI
tff(fact_889_subsetI,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( ! [X5: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X5),A3))
         => pp(aa_set_set_nat_bool(member_set_nat(X5),B2)) )
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),B2)) ) ).

% subsetI
tff(fact_890_subsetI,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),A3))
         => pp(aa_set_nat_bool(member_nat(X5),B2)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2)) ) ).

% subsetI
tff(fact_891_subsetI,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),A3))
         => pp(aa_set_int_bool(member_int(X5),B2)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2)) ) ).

% subsetI
tff(fact_892_subset__antisym,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A3))
       => ( A3 = B2 ) ) ) ).

% subset_antisym
tff(fact_893_psubsetI,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
     => ( ( A3 != B2 )
       => pp(aa_set_int_bool(ord_less_set_int(A3),B2)) ) ) ).

% psubsetI
tff(fact_894_in__mono,axiom,
    ! [A3: set_real,B2: set_real,X: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2))
     => ( pp(aa_set_real_bool(member_real(X),A3))
       => pp(aa_set_real_bool(member_real(X),B2)) ) ) ).

% in_mono
tff(fact_895_in__mono,axiom,
    ! [A3: set_bool,B2: set_bool,X: bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),B2))
     => ( pp(aa_set_bool_bool(member_bool(X),A3))
       => pp(aa_set_bool_bool(member_bool(X),B2)) ) ) ).

% in_mono
tff(fact_896_in__mono,axiom,
    ! [A3: set_set_nat,B2: set_set_nat,X: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),B2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X),A3))
       => pp(aa_set_set_nat_bool(member_set_nat(X),B2)) ) ) ).

% in_mono
tff(fact_897_in__mono,axiom,
    ! [A3: set_nat,B2: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
     => ( pp(aa_set_nat_bool(member_nat(X),A3))
       => pp(aa_set_nat_bool(member_nat(X),B2)) ) ) ).

% in_mono
tff(fact_898_in__mono,axiom,
    ! [A3: set_int,B2: set_int,X: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
     => ( pp(aa_set_int_bool(member_int(X),A3))
       => pp(aa_set_int_bool(member_int(X),B2)) ) ) ).

% in_mono
tff(fact_899_subsetD,axiom,
    ! [A3: set_real,B2: set_real,C: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2))
     => ( pp(aa_set_real_bool(member_real(C),A3))
       => pp(aa_set_real_bool(member_real(C),B2)) ) ) ).

% subsetD
tff(fact_900_subsetD,axiom,
    ! [A3: set_bool,B2: set_bool,C: bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),B2))
     => ( pp(aa_set_bool_bool(member_bool(C),A3))
       => pp(aa_set_bool_bool(member_bool(C),B2)) ) ) ).

% subsetD
tff(fact_901_subsetD,axiom,
    ! [A3: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),B2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(C),A3))
       => pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ) ).

% subsetD
tff(fact_902_subsetD,axiom,
    ! [A3: set_nat,B2: set_nat,C: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
     => ( pp(aa_set_nat_bool(member_nat(C),A3))
       => pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).

% subsetD
tff(fact_903_subsetD,axiom,
    ! [A3: set_int,B2: set_int,C: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
     => ( pp(aa_set_int_bool(member_int(C),A3))
       => pp(aa_set_int_bool(member_int(C),B2)) ) ) ).

% subsetD
tff(fact_904_psubsetD,axiom,
    ! [A3: set_real,B2: set_real,C: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A3),B2))
     => ( pp(aa_set_real_bool(member_real(C),A3))
       => pp(aa_set_real_bool(member_real(C),B2)) ) ) ).

% psubsetD
tff(fact_905_psubsetD,axiom,
    ! [A3: set_bool,B2: set_bool,C: bool] :
      ( pp(aa_set_bool_bool(ord_less_set_bool(A3),B2))
     => ( pp(aa_set_bool_bool(member_bool(C),A3))
       => pp(aa_set_bool_bool(member_bool(C),B2)) ) ) ).

% psubsetD
tff(fact_906_psubsetD,axiom,
    ! [A3: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A3),B2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(C),A3))
       => pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ) ).

% psubsetD
tff(fact_907_psubsetD,axiom,
    ! [A3: set_nat,B2: set_nat,C: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A3),B2))
     => ( pp(aa_set_nat_bool(member_nat(C),A3))
       => pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).

% psubsetD
tff(fact_908_psubsetD,axiom,
    ! [A3: set_int,B2: set_int,C: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A3),B2))
     => ( pp(aa_set_int_bool(member_int(C),A3))
       => pp(aa_set_int_bool(member_int(C),B2)) ) ) ).

% psubsetD
tff(fact_909_psubsetE,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A3),B2))
     => ~ ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(B2),A3)) ) ) ).

% psubsetE
tff(fact_910_equalityE,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( A3 = B2 )
     => ~ ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
         => ~ pp(aa_set_int_bool(ord_less_eq_set_int(B2),A3)) ) ) ).

% equalityE
tff(fact_911_subset__eq,axiom,
    ! [A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2))
    <=> ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A3))
         => pp(aa_set_real_bool(member_real(X3),B2)) ) ) ).

% subset_eq
tff(fact_912_subset__eq,axiom,
    ! [A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),B2))
    <=> ! [X3: bool] :
          ( pp(aa_set_bool_bool(member_bool(X3),A3))
         => pp(aa_set_bool_bool(member_bool(X3),B2)) ) ) ).

% subset_eq
tff(fact_913_subset__eq,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),B2))
    <=> ! [X3: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X3),A3))
         => pp(aa_set_set_nat_bool(member_set_nat(X3),B2)) ) ) ).

% subset_eq
tff(fact_914_subset__eq,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A3))
         => pp(aa_set_nat_bool(member_nat(X3),B2)) ) ) ).

% subset_eq
tff(fact_915_subset__eq,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
    <=> ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A3))
         => pp(aa_set_int_bool(member_int(X3),B2)) ) ) ).

% subset_eq
tff(fact_916_equalityD1,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( A3 = B2 )
     => pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2)) ) ).

% equalityD1
tff(fact_917_equalityD2,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( A3 = B2 )
     => pp(aa_set_int_bool(ord_less_eq_set_int(B2),A3)) ) ).

% equalityD2
tff(fact_918_psubset__eq,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A3),B2))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
        & ( A3 != B2 ) ) ) ).

% psubset_eq
tff(fact_919_subset__iff,axiom,
    ! [A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2))
    <=> ! [T3: real] :
          ( pp(aa_set_real_bool(member_real(T3),A3))
         => pp(aa_set_real_bool(member_real(T3),B2)) ) ) ).

% subset_iff
tff(fact_920_subset__iff,axiom,
    ! [A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),B2))
    <=> ! [T3: bool] :
          ( pp(aa_set_bool_bool(member_bool(T3),A3))
         => pp(aa_set_bool_bool(member_bool(T3),B2)) ) ) ).

% subset_iff
tff(fact_921_subset__iff,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),B2))
    <=> ! [T3: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(T3),A3))
         => pp(aa_set_set_nat_bool(member_set_nat(T3),B2)) ) ) ).

% subset_iff
tff(fact_922_subset__iff,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
    <=> ! [T3: nat] :
          ( pp(aa_set_nat_bool(member_nat(T3),A3))
         => pp(aa_set_nat_bool(member_nat(T3),B2)) ) ) ).

% subset_iff
tff(fact_923_subset__iff,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
    <=> ! [T3: int] :
          ( pp(aa_set_int_bool(member_int(T3),A3))
         => pp(aa_set_int_bool(member_int(T3),B2)) ) ) ).

% subset_iff
tff(fact_924_subset__refl,axiom,
    ! [A3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A3),A3)) ).

% subset_refl
tff(fact_925_Collect__mono,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( ! [X5: real] :
          ( pp(aa_real_bool(P,X5))
         => pp(aa_real_bool(Q,X5)) )
     => pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(P)),collect_real(Q))) ) ).

% Collect_mono
tff(fact_926_Collect__mono,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ! [X5: list_nat] :
          ( pp(aa_list_nat_bool(P,X5))
         => pp(aa_list_nat_bool(Q,X5)) )
     => ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q)) ) ).

% Collect_mono
tff(fact_927_Collect__mono,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ! [X5: set_nat] :
          ( pp(aa_set_nat_bool(P,X5))
         => pp(aa_set_nat_bool(Q,X5)) )
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q))) ) ).

% Collect_mono
tff(fact_928_Collect__mono,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X5: nat] :
          ( pp(aa_nat_bool(P,X5))
         => pp(aa_nat_bool(Q,X5)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(P)),collect_nat(Q))) ) ).

% Collect_mono
tff(fact_929_Collect__mono,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X5: int] :
          ( pp(aa_int_bool(P,X5))
         => pp(aa_int_bool(Q,X5)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(P)),collect_int(Q))) ) ).

% Collect_mono
tff(fact_930_subset__trans,axiom,
    ! [A3: set_int,B2: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A3),C2)) ) ) ).

% subset_trans
tff(fact_931_set__eq__subset,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( A3 = B2 )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
        & pp(aa_set_int_bool(ord_less_eq_set_int(B2),A3)) ) ) ).

% set_eq_subset
tff(fact_932_Collect__mono__iff,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(P)),collect_real(Q)))
    <=> ! [X3: real] :
          ( pp(aa_real_bool(P,X3))
         => pp(aa_real_bool(Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_933_Collect__mono__iff,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q))
    <=> ! [X3: list_nat] :
          ( pp(aa_list_nat_bool(P,X3))
         => pp(aa_list_nat_bool(Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_934_Collect__mono__iff,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q)))
    <=> ! [X3: set_nat] :
          ( pp(aa_set_nat_bool(P,X3))
         => pp(aa_set_nat_bool(Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_935_Collect__mono__iff,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(P)),collect_nat(Q)))
    <=> ! [X3: nat] :
          ( pp(aa_nat_bool(P,X3))
         => pp(aa_nat_bool(Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_936_Collect__mono__iff,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(P)),collect_int(Q)))
    <=> ! [X3: int] :
          ( pp(aa_int_bool(P,X3))
         => pp(aa_int_bool(Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_937_psubset__imp__subset,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A3),B2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2)) ) ).

% psubset_imp_subset
tff(fact_938_psubset__subset__trans,axiom,
    ! [A3: set_int,B2: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A3),B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C2))
       => pp(aa_set_int_bool(ord_less_set_int(A3),C2)) ) ) ).

% psubset_subset_trans
tff(fact_939_subset__not__subset__eq,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A3),B2))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(B2),A3)) ) ) ).

% subset_not_subset_eq
tff(fact_940_subset__psubset__trans,axiom,
    ! [A3: set_int,B2: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
     => ( pp(aa_set_int_bool(ord_less_set_int(B2),C2))
       => pp(aa_set_int_bool(ord_less_set_int(A3),C2)) ) ) ).

% subset_psubset_trans
tff(fact_941_subset__iff__psubset__eq,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(A3),B2))
        | ( A3 = B2 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_942_bounded__Max__nat,axiom,
    ! [P: fun_nat_bool,X: nat,M5: nat] :
      ( pp(aa_nat_bool(P,X))
     => ( ! [X5: nat] :
            ( pp(aa_nat_bool(P,X5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),M5)) )
       => ~ ! [M4: nat] :
              ( pp(aa_nat_bool(P,M4))
             => ~ ! [X4: nat] :
                    ( pp(aa_nat_bool(P,X4))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),M4)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_943_finite__transitivity__chain,axiom,
    ! [A3: set_set_nat,R: fun_se707272970763642173t_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( ! [X5: set_nat] : ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X5),X5))
       => ( ! [X5: set_nat,Y3: set_nat,Z3: set_nat] :
              ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X5),Y3))
             => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,Y3),Z3))
               => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X5),Z3)) ) )
         => ( ! [X5: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(X5),A3))
               => ? [Y4: set_nat] :
                    ( pp(aa_set_set_nat_bool(member_set_nat(Y4),A3))
                    & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X5),Y4)) ) )
           => ( A3 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_944_finite__transitivity__chain,axiom,
    ! [A3: set_complex,R: fun_co1378794544173401837x_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ! [X5: complex] : ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X5),X5))
       => ( ! [X5: complex,Y3: complex,Z3: complex] :
              ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X5),Y3))
             => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y3),Z3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X5),Z3)) ) )
         => ( ! [X5: complex] :
                ( pp(aa_set_complex_bool(member_complex(X5),A3))
               => ? [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),A3))
                    & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X5),Y4)) ) )
           => ( A3 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_945_finite__transitivity__chain,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: fun_Pr248683899324676269t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( ! [X5: product_prod_nat_nat] : ~ pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,X5),X5))
       => ( ! [X5: product_prod_nat_nat,Y3: product_prod_nat_nat,Z3: product_prod_nat_nat] :
              ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,X5),Y3))
             => ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,Y3),Z3))
               => pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,X5),Z3)) ) )
         => ( ! [X5: product_prod_nat_nat] :
                ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),A3))
               => ? [Y4: product_prod_nat_nat] :
                    ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y4),A3))
                    & pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,X5),Y4)) ) )
           => ( A3 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_946_finite__transitivity__chain,axiom,
    ! [A3: set_Extended_enat,R: fun_Ex8598408993869023341t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ! [X5: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(R,X5),X5))
       => ( ! [X5: extended_enat,Y3: extended_enat,Z3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(R,X5),Y3))
             => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(R,Y3),Z3))
               => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(R,X5),Z3)) ) )
         => ( ! [X5: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A3))
               => ? [Y4: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y4),A3))
                    & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(R,X5),Y4)) ) )
           => ( A3 = bot_bo7653980558646680370d_enat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_947_finite__transitivity__chain,axiom,
    ! [A3: set_real,R: fun_re1255401033344146673l_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ! [X5: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X5),X5))
       => ( ! [X5: real,Y3: real,Z3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X5),Y3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y3),Z3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X5),Z3)) ) )
         => ( ! [X5: real] :
                ( pp(aa_set_real_bool(member_real(X5),A3))
               => ? [Y4: real] :
                    ( pp(aa_set_real_bool(member_real(Y4),A3))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X5),Y4)) ) )
           => ( A3 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_948_finite__transitivity__chain,axiom,
    ! [A3: set_bool,R: fun_bo1841576699660343469l_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( ! [X5: bool] : ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(R,X5),X5))
       => ( ! [X5: bool,Y3: bool,Z3: bool] :
              ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(R,X5),Y3))
             => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(R,Y3),Z3))
               => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(R,X5),Z3)) ) )
         => ( ! [X5: bool] :
                ( pp(aa_set_bool_bool(member_bool(X5),A3))
               => ? [Y4: bool] :
                    ( pp(aa_set_bool_bool(member_bool(Y4),A3))
                    & pp(aa_bool_bool(aa_boo8970175072276269406l_bool(R,X5),Y4)) ) )
           => ( A3 = bot_bot_set_bool ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_949_finite__transitivity__chain,axiom,
    ! [A3: set_nat,R: fun_nat_fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ! [X5: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X5),X5))
       => ( ! [X5: nat,Y3: nat,Z3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X5),Y3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y3),Z3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X5),Z3)) ) )
         => ( ! [X5: nat] :
                ( pp(aa_set_nat_bool(member_nat(X5),A3))
               => ? [Y4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y4),A3))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X5),Y4)) ) )
           => ( A3 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_950_finite__transitivity__chain,axiom,
    ! [A3: set_int,R: fun_int_fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ! [X5: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(R,X5),X5))
       => ( ! [X5: int,Y3: int,Z3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(R,X5),Y3))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(R,Y3),Z3))
               => pp(aa_int_bool(aa_int_fun_int_bool(R,X5),Z3)) ) )
         => ( ! [X5: int] :
                ( pp(aa_set_int_bool(member_int(X5),A3))
               => ? [Y4: int] :
                    ( pp(aa_set_int_bool(member_int(Y4),A3))
                    & pp(aa_int_bool(aa_int_fun_int_bool(R,X5),Y4)) ) )
           => ( A3 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_951_bot__empty__eq,axiom,
    ! [X4: set_nat] :
      ( pp(aa_set_nat_bool(bot_bo8343409700046576086t_bool,X4))
    <=> pp(aa_set_set_nat_bool(member_set_nat(X4),bot_bot_set_set_nat)) ) ).

% bot_empty_eq
tff(fact_952_bot__empty__eq,axiom,
    ! [X4: real] :
      ( pp(aa_real_bool(bot_bo4181092155253064880l_bool,X4))
    <=> pp(aa_set_real_bool(member_real(X4),bot_bot_set_real)) ) ).

% bot_empty_eq
tff(fact_953_bot__empty__eq,axiom,
    ! [X4: bool] :
      ( pp(aa_bool_bool(bot_bo4756488196683355415l_bool,X4))
    <=> pp(aa_set_bool_bool(member_bool(X4),bot_bot_set_bool)) ) ).

% bot_empty_eq
tff(fact_954_bot__empty__eq,axiom,
    ! [X4: nat] :
      ( pp(aa_nat_bool(bot_bot_fun_nat_bool,X4))
    <=> pp(aa_set_nat_bool(member_nat(X4),bot_bot_set_nat)) ) ).

% bot_empty_eq
tff(fact_955_bot__empty__eq,axiom,
    ! [X4: int] :
      ( pp(aa_int_bool(bot_bot_fun_int_bool,X4))
    <=> pp(aa_set_int_bool(member_int(X4),bot_bot_set_int)) ) ).

% bot_empty_eq
tff(fact_956_Collect__empty__eq__bot,axiom,
    ! [P: fun_list_nat_bool] :
      ( ( collect_list_nat(P) = bot_bot_set_list_nat )
    <=> ( P = bot_bo6469247547558333820t_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_957_Collect__empty__eq__bot,axiom,
    ! [P: fun_set_nat_bool] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ( P = bot_bo8343409700046576086t_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_958_Collect__empty__eq__bot,axiom,
    ! [P: fun_real_bool] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ( P = bot_bo4181092155253064880l_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_959_Collect__empty__eq__bot,axiom,
    ! [P: fun_bool_bool] :
      ( ( collect_bool(P) = bot_bot_set_bool )
    <=> ( P = bot_bo4756488196683355415l_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_960_Collect__empty__eq__bot,axiom,
    ! [P: fun_nat_bool] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ( P = bot_bot_fun_nat_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_961_Collect__empty__eq__bot,axiom,
    ! [P: fun_int_bool] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ( P = bot_bot_fun_int_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_962_subset__emptyI,axiom,
    ! [A3: set_set_nat] :
      ( ! [X5: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(X5),A3))
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),bot_bot_set_set_nat)) ) ).

% subset_emptyI
tff(fact_963_subset__emptyI,axiom,
    ! [A3: set_real] :
      ( ! [X5: real] : ~ pp(aa_set_real_bool(member_real(X5),A3))
     => pp(aa_set_real_bool(ord_less_eq_set_real(A3),bot_bot_set_real)) ) ).

% subset_emptyI
tff(fact_964_subset__emptyI,axiom,
    ! [A3: set_bool] :
      ( ! [X5: bool] : ~ pp(aa_set_bool_bool(member_bool(X5),A3))
     => pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),bot_bot_set_bool)) ) ).

% subset_emptyI
tff(fact_965_subset__emptyI,axiom,
    ! [A3: set_nat] :
      ( ! [X5: nat] : ~ pp(aa_set_nat_bool(member_nat(X5),A3))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),bot_bot_set_nat)) ) ).

% subset_emptyI
tff(fact_966_subset__emptyI,axiom,
    ! [A3: set_int] :
      ( ! [X5: int] : ~ pp(aa_set_int_bool(member_int(X5),A3))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A3),bot_bot_set_int)) ) ).

% subset_emptyI
tff(fact_967_field__lbound__gt__zero,axiom,
    ! [D1: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
       => ? [E: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E),D1))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E),D2)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_968_field__lbound__gt__zero,axiom,
    ! [D1: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D1))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D2))
       => ? [E: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E),D1))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E),D2)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_969_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),zero_zero_real)) ).

% less_numeral_extra(3)
tff(fact_970_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),zero_zero_rat)) ).

% less_numeral_extra(3)
tff(fact_971_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),zero_zero_nat)) ).

% less_numeral_extra(3)
tff(fact_972_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).

% less_numeral_extra(3)
tff(fact_973_complete__interval,axiom,
    ! [A2: real,B3: real,P: fun_real_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(P,A2))
       => ( ~ pp(aa_real_bool(P,B3))
         => ? [C3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),C3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C3),B3))
              & ! [X4: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X4))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),C3)) )
                 => pp(aa_real_bool(P,X4)) )
              & ! [D3: real] :
                  ( ! [X5: real] :
                      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X5))
                        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),D3)) )
                     => pp(aa_real_bool(P,X5)) )
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D3),C3)) ) ) ) ) ) ).

% complete_interval
tff(fact_974_complete__interval,axiom,
    ! [A2: nat,B3: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( pp(aa_nat_bool(P,A2))
       => ( ~ pp(aa_nat_bool(P,B3))
         => ? [C3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C3))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C3),B3))
              & ! [X4: nat] :
                  ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),X4))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),C3)) )
                 => pp(aa_nat_bool(P,X4)) )
              & ! [D3: nat] :
                  ( ! [X5: nat] :
                      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),X5))
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),D3)) )
                     => pp(aa_nat_bool(P,X5)) )
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D3),C3)) ) ) ) ) ) ).

% complete_interval
tff(fact_975_complete__interval,axiom,
    ! [A2: int,B3: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( pp(aa_int_bool(P,A2))
       => ( ~ pp(aa_int_bool(P,B3))
         => ? [C3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C3),B3))
              & ! [X4: int] :
                  ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),X4))
                    & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),C3)) )
                 => pp(aa_int_bool(P,X4)) )
              & ! [D3: int] :
                  ( ! [X5: int] :
                      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),X5))
                        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),D3)) )
                     => pp(aa_int_bool(P,X5)) )
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D3),C3)) ) ) ) ) ) ).

% complete_interval
tff(fact_976_verit__comp__simplify1_I3_J,axiom,
    ! [B5: real,A5: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B5),A5))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_977_verit__comp__simplify1_I3_J,axiom,
    ! [B5: rat,A5: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B5),A5))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_978_verit__comp__simplify1_I3_J,axiom,
    ! [B5: num,A5: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B5),A5))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_979_verit__comp__simplify1_I3_J,axiom,
    ! [B5: nat,A5: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B5),A5))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_980_verit__comp__simplify1_I3_J,axiom,
    ! [B5: int,A5: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),A5))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_981_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),T)) ) ).

% pinf(6)
tff(fact_982_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),T)) ) ).

% pinf(6)
tff(fact_983_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),T)) ) ).

% pinf(6)
tff(fact_984_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),T)) ) ).

% pinf(6)
tff(fact_985_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),T)) ) ).

% pinf(6)
tff(fact_986_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X4)) ) ).

% pinf(8)
tff(fact_987_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X4)) ) ).

% pinf(8)
tff(fact_988_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X4)) ) ).

% pinf(8)
tff(fact_989_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X4)) ) ).

% pinf(8)
tff(fact_990_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X4)) ) ).

% pinf(8)
tff(fact_991_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),T)) ) ).

% minf(6)
tff(fact_992_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),T)) ) ).

% minf(6)
tff(fact_993_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),T)) ) ).

% minf(6)
tff(fact_994_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),T)) ) ).

% minf(6)
tff(fact_995_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),T)) ) ).

% minf(6)
tff(fact_996_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X4)) ) ).

% minf(8)
tff(fact_997_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X4)) ) ).

% minf(8)
tff(fact_998_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X4)) ) ).

% minf(8)
tff(fact_999_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X4)) ) ).

% minf(8)
tff(fact_1000_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X4)) ) ).

% minf(8)
tff(fact_1001_verit__la__disequality,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 = B3 )
      | ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
      | ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2)) ) ).

% verit_la_disequality
tff(fact_1002_verit__la__disequality,axiom,
    ! [A2: num,B3: num] :
      ( ( A2 = B3 )
      | ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
      | ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2)) ) ).

% verit_la_disequality
tff(fact_1003_verit__la__disequality,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 = B3 )
      | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
      | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2)) ) ).

% verit_la_disequality
tff(fact_1004_verit__la__disequality,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 = B3 )
      | ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
      | ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2)) ) ).

% verit_la_disequality
tff(fact_1005_verit__comp__simplify1_I2_J,axiom,
    ! [A2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A2),A2)) ).

% verit_comp_simplify1(2)
tff(fact_1006_verit__comp__simplify1_I2_J,axiom,
    ! [A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),A2)) ).

% verit_comp_simplify1(2)
tff(fact_1007_verit__comp__simplify1_I2_J,axiom,
    ! [A2: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),A2)) ).

% verit_comp_simplify1(2)
tff(fact_1008_verit__comp__simplify1_I2_J,axiom,
    ! [A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),A2)) ).

% verit_comp_simplify1(2)
tff(fact_1009_verit__comp__simplify1_I2_J,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),A2)) ).

% verit_comp_simplify1(2)
tff(fact_1010_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X4)) ) ).

% minf(7)
tff(fact_1011_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X4)) ) ).

% minf(7)
tff(fact_1012_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X4)) ) ).

% minf(7)
tff(fact_1013_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X4)) ) ).

% minf(7)
tff(fact_1014_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X4)) ) ).

% minf(7)
tff(fact_1015_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),T)) ) ).

% minf(5)
tff(fact_1016_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),T)) ) ).

% minf(5)
tff(fact_1017_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),T)) ) ).

% minf(5)
tff(fact_1018_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),T)) ) ).

% minf(5)
tff(fact_1019_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),T)) ) ).

% minf(5)
tff(fact_1020_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_1021_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_1022_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_1023_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_1024_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_1025_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_1026_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_1027_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_1028_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_1029_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_1030_minf_I2_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q2: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z4))
         => ( pp(aa_real_bool(P,X5))
          <=> pp(aa_real_bool(P2,X5)) ) )
     => ( ? [Z4: real] :
          ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z4))
           => ( pp(aa_real_bool(Q,X5))
            <=> pp(aa_real_bool(Q2,X5)) ) )
       => ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
           => ( ( pp(aa_real_bool(P,X4))
                | pp(aa_real_bool(Q,X4)) )
            <=> ( pp(aa_real_bool(P2,X4))
                | pp(aa_real_bool(Q2,X4)) ) ) ) ) ) ).

% minf(2)
tff(fact_1031_minf_I2_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q2: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z4))
         => ( pp(aa_rat_bool(P,X5))
          <=> pp(aa_rat_bool(P2,X5)) ) )
     => ( ? [Z4: rat] :
          ! [X5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z4))
           => ( pp(aa_rat_bool(Q,X5))
            <=> pp(aa_rat_bool(Q2,X5)) ) )
       => ? [Z3: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
           => ( ( pp(aa_rat_bool(P,X4))
                | pp(aa_rat_bool(Q,X4)) )
            <=> ( pp(aa_rat_bool(P2,X4))
                | pp(aa_rat_bool(Q2,X4)) ) ) ) ) ) ).

% minf(2)
tff(fact_1032_minf_I2_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X5: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z4))
         => ( pp(aa_num_bool(P,X5))
          <=> pp(aa_num_bool(P2,X5)) ) )
     => ( ? [Z4: num] :
          ! [X5: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z4))
           => ( pp(aa_num_bool(Q,X5))
            <=> pp(aa_num_bool(Q2,X5)) ) )
       => ? [Z3: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
           => ( ( pp(aa_num_bool(P,X4))
                | pp(aa_num_bool(Q,X4)) )
            <=> ( pp(aa_num_bool(P2,X4))
                | pp(aa_num_bool(Q2,X4)) ) ) ) ) ) ).

% minf(2)
tff(fact_1033_minf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z4))
         => ( pp(aa_nat_bool(P,X5))
          <=> pp(aa_nat_bool(P2,X5)) ) )
     => ( ? [Z4: nat] :
          ! [X5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z4))
           => ( pp(aa_nat_bool(Q,X5))
            <=> pp(aa_nat_bool(Q2,X5)) ) )
       => ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
           => ( ( pp(aa_nat_bool(P,X4))
                | pp(aa_nat_bool(Q,X4)) )
            <=> ( pp(aa_nat_bool(P2,X4))
                | pp(aa_nat_bool(Q2,X4)) ) ) ) ) ) ).

% minf(2)
tff(fact_1034_minf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z4))
         => ( pp(aa_int_bool(P,X5))
          <=> pp(aa_int_bool(P2,X5)) ) )
     => ( ? [Z4: int] :
          ! [X5: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z4))
           => ( pp(aa_int_bool(Q,X5))
            <=> pp(aa_int_bool(Q2,X5)) ) )
       => ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
           => ( ( pp(aa_int_bool(P,X4))
                | pp(aa_int_bool(Q,X4)) )
            <=> ( pp(aa_int_bool(P2,X4))
                | pp(aa_int_bool(Q2,X4)) ) ) ) ) ) ).

% minf(2)
tff(fact_1035_minf_I1_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q2: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z4))
         => ( pp(aa_real_bool(P,X5))
          <=> pp(aa_real_bool(P2,X5)) ) )
     => ( ? [Z4: real] :
          ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z4))
           => ( pp(aa_real_bool(Q,X5))
            <=> pp(aa_real_bool(Q2,X5)) ) )
       => ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
           => ( ( pp(aa_real_bool(P,X4))
                & pp(aa_real_bool(Q,X4)) )
            <=> ( pp(aa_real_bool(P2,X4))
                & pp(aa_real_bool(Q2,X4)) ) ) ) ) ) ).

% minf(1)
tff(fact_1036_minf_I1_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q2: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z4))
         => ( pp(aa_rat_bool(P,X5))
          <=> pp(aa_rat_bool(P2,X5)) ) )
     => ( ? [Z4: rat] :
          ! [X5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z4))
           => ( pp(aa_rat_bool(Q,X5))
            <=> pp(aa_rat_bool(Q2,X5)) ) )
       => ? [Z3: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
           => ( ( pp(aa_rat_bool(P,X4))
                & pp(aa_rat_bool(Q,X4)) )
            <=> ( pp(aa_rat_bool(P2,X4))
                & pp(aa_rat_bool(Q2,X4)) ) ) ) ) ) ).

% minf(1)
tff(fact_1037_minf_I1_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X5: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z4))
         => ( pp(aa_num_bool(P,X5))
          <=> pp(aa_num_bool(P2,X5)) ) )
     => ( ? [Z4: num] :
          ! [X5: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z4))
           => ( pp(aa_num_bool(Q,X5))
            <=> pp(aa_num_bool(Q2,X5)) ) )
       => ? [Z3: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
           => ( ( pp(aa_num_bool(P,X4))
                & pp(aa_num_bool(Q,X4)) )
            <=> ( pp(aa_num_bool(P2,X4))
                & pp(aa_num_bool(Q2,X4)) ) ) ) ) ) ).

% minf(1)
tff(fact_1038_minf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z4))
         => ( pp(aa_nat_bool(P,X5))
          <=> pp(aa_nat_bool(P2,X5)) ) )
     => ( ? [Z4: nat] :
          ! [X5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z4))
           => ( pp(aa_nat_bool(Q,X5))
            <=> pp(aa_nat_bool(Q2,X5)) ) )
       => ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
           => ( ( pp(aa_nat_bool(P,X4))
                & pp(aa_nat_bool(Q,X4)) )
            <=> ( pp(aa_nat_bool(P2,X4))
                & pp(aa_nat_bool(Q2,X4)) ) ) ) ) ) ).

% minf(1)
tff(fact_1039_minf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z4))
         => ( pp(aa_int_bool(P,X5))
          <=> pp(aa_int_bool(P2,X5)) ) )
     => ( ? [Z4: int] :
          ! [X5: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z4))
           => ( pp(aa_int_bool(Q,X5))
            <=> pp(aa_int_bool(Q2,X5)) ) )
       => ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
           => ( ( pp(aa_int_bool(P,X4))
                & pp(aa_int_bool(Q,X4)) )
            <=> ( pp(aa_int_bool(P2,X4))
                & pp(aa_int_bool(Q2,X4)) ) ) ) ) ) ).

% minf(1)
tff(fact_1040_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X4)) ) ).

% pinf(7)
tff(fact_1041_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X4)) ) ).

% pinf(7)
tff(fact_1042_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X4)) ) ).

% pinf(7)
tff(fact_1043_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X4)) ) ).

% pinf(7)
tff(fact_1044_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X4)) ) ).

% pinf(7)
tff(fact_1045_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),T)) ) ).

% pinf(5)
tff(fact_1046_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),T)) ) ).

% pinf(5)
tff(fact_1047_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),T)) ) ).

% pinf(5)
tff(fact_1048_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),T)) ) ).

% pinf(5)
tff(fact_1049_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),T)) ) ).

% pinf(5)
tff(fact_1050_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_1051_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_1052_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_1053_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_1054_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_1055_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_1056_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_1057_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_1058_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_1059_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_1060_pinf_I2_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q2: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X5))
         => ( pp(aa_real_bool(P,X5))
          <=> pp(aa_real_bool(P2,X5)) ) )
     => ( ? [Z4: real] :
          ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X5))
           => ( pp(aa_real_bool(Q,X5))
            <=> pp(aa_real_bool(Q2,X5)) ) )
       => ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
           => ( ( pp(aa_real_bool(P,X4))
                | pp(aa_real_bool(Q,X4)) )
            <=> ( pp(aa_real_bool(P2,X4))
                | pp(aa_real_bool(Q2,X4)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1061_pinf_I2_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q2: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X5))
         => ( pp(aa_rat_bool(P,X5))
          <=> pp(aa_rat_bool(P2,X5)) ) )
     => ( ? [Z4: rat] :
          ! [X5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X5))
           => ( pp(aa_rat_bool(Q,X5))
            <=> pp(aa_rat_bool(Q2,X5)) ) )
       => ? [Z3: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
           => ( ( pp(aa_rat_bool(P,X4))
                | pp(aa_rat_bool(Q,X4)) )
            <=> ( pp(aa_rat_bool(P2,X4))
                | pp(aa_rat_bool(Q2,X4)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1062_pinf_I2_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X5: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X5))
         => ( pp(aa_num_bool(P,X5))
          <=> pp(aa_num_bool(P2,X5)) ) )
     => ( ? [Z4: num] :
          ! [X5: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X5))
           => ( pp(aa_num_bool(Q,X5))
            <=> pp(aa_num_bool(Q2,X5)) ) )
       => ? [Z3: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
           => ( ( pp(aa_num_bool(P,X4))
                | pp(aa_num_bool(Q,X4)) )
            <=> ( pp(aa_num_bool(P2,X4))
                | pp(aa_num_bool(Q2,X4)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1063_pinf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X5))
         => ( pp(aa_nat_bool(P,X5))
          <=> pp(aa_nat_bool(P2,X5)) ) )
     => ( ? [Z4: nat] :
          ! [X5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X5))
           => ( pp(aa_nat_bool(Q,X5))
            <=> pp(aa_nat_bool(Q2,X5)) ) )
       => ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
           => ( ( pp(aa_nat_bool(P,X4))
                | pp(aa_nat_bool(Q,X4)) )
            <=> ( pp(aa_nat_bool(P2,X4))
                | pp(aa_nat_bool(Q2,X4)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1064_pinf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X5))
         => ( pp(aa_int_bool(P,X5))
          <=> pp(aa_int_bool(P2,X5)) ) )
     => ( ? [Z4: int] :
          ! [X5: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X5))
           => ( pp(aa_int_bool(Q,X5))
            <=> pp(aa_int_bool(Q2,X5)) ) )
       => ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
           => ( ( pp(aa_int_bool(P,X4))
                | pp(aa_int_bool(Q,X4)) )
            <=> ( pp(aa_int_bool(P2,X4))
                | pp(aa_int_bool(Q2,X4)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1065_pinf_I1_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q2: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X5))
         => ( pp(aa_real_bool(P,X5))
          <=> pp(aa_real_bool(P2,X5)) ) )
     => ( ? [Z4: real] :
          ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X5))
           => ( pp(aa_real_bool(Q,X5))
            <=> pp(aa_real_bool(Q2,X5)) ) )
       => ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
           => ( ( pp(aa_real_bool(P,X4))
                & pp(aa_real_bool(Q,X4)) )
            <=> ( pp(aa_real_bool(P2,X4))
                & pp(aa_real_bool(Q2,X4)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1066_pinf_I1_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q2: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X5))
         => ( pp(aa_rat_bool(P,X5))
          <=> pp(aa_rat_bool(P2,X5)) ) )
     => ( ? [Z4: rat] :
          ! [X5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X5))
           => ( pp(aa_rat_bool(Q,X5))
            <=> pp(aa_rat_bool(Q2,X5)) ) )
       => ? [Z3: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
           => ( ( pp(aa_rat_bool(P,X4))
                & pp(aa_rat_bool(Q,X4)) )
            <=> ( pp(aa_rat_bool(P2,X4))
                & pp(aa_rat_bool(Q2,X4)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1067_pinf_I1_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X5: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X5))
         => ( pp(aa_num_bool(P,X5))
          <=> pp(aa_num_bool(P2,X5)) ) )
     => ( ? [Z4: num] :
          ! [X5: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X5))
           => ( pp(aa_num_bool(Q,X5))
            <=> pp(aa_num_bool(Q2,X5)) ) )
       => ? [Z3: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
           => ( ( pp(aa_num_bool(P,X4))
                & pp(aa_num_bool(Q,X4)) )
            <=> ( pp(aa_num_bool(P2,X4))
                & pp(aa_num_bool(Q2,X4)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1068_pinf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X5))
         => ( pp(aa_nat_bool(P,X5))
          <=> pp(aa_nat_bool(P2,X5)) ) )
     => ( ? [Z4: nat] :
          ! [X5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X5))
           => ( pp(aa_nat_bool(Q,X5))
            <=> pp(aa_nat_bool(Q2,X5)) ) )
       => ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
           => ( ( pp(aa_nat_bool(P,X4))
                & pp(aa_nat_bool(Q,X4)) )
            <=> ( pp(aa_nat_bool(P2,X4))
                & pp(aa_nat_bool(Q2,X4)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1069_pinf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X5))
         => ( pp(aa_int_bool(P,X5))
          <=> pp(aa_int_bool(P2,X5)) ) )
     => ( ? [Z4: int] :
          ! [X5: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X5))
           => ( pp(aa_int_bool(Q,X5))
            <=> pp(aa_int_bool(Q2,X5)) ) )
       => ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
           => ( ( pp(aa_int_bool(P,X4))
                & pp(aa_int_bool(Q,X4)) )
            <=> ( pp(aa_int_bool(P2,X4))
                & pp(aa_int_bool(Q2,X4)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1070_verit__comp__simplify1_I1_J,axiom,
    ! [A2: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),A2)) ).

% verit_comp_simplify1(1)
tff(fact_1071_verit__comp__simplify1_I1_J,axiom,
    ! [A2: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),A2)) ).

% verit_comp_simplify1(1)
tff(fact_1072_verit__comp__simplify1_I1_J,axiom,
    ! [A2: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),A2)) ).

% verit_comp_simplify1(1)
tff(fact_1073_verit__comp__simplify1_I1_J,axiom,
    ! [A2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),A2)) ).

% verit_comp_simplify1(1)
tff(fact_1074_verit__comp__simplify1_I1_J,axiom,
    ! [A2: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),A2)) ).

% verit_comp_simplify1(1)
tff(fact_1075_ex__gt__or__lt,axiom,
    ! [A2: real] :
    ? [B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A2)) ) ).

% ex_gt_or_lt
tff(fact_1076_le__numeral__extra_I3_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),zero_zero_real)) ).

% le_numeral_extra(3)
tff(fact_1077_le__numeral__extra_I3_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),zero_zero_rat)) ).

% le_numeral_extra(3)
tff(fact_1078_le__numeral__extra_I3_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),zero_zero_nat)) ).

% le_numeral_extra(3)
tff(fact_1079_le__numeral__extra_I3_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

% le_numeral_extra(3)
tff(fact_1080_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
    <=> ( pp(aa_nat_bool(vEBT_V5719532721284313246member(T),X))
        | pp(aa_nat_bool(vEBT_VEBT_membermima(T),X)) ) ) ).

% both_member_options_def
tff(fact_1081_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
       => pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ) ).

% valid_member_both_member_options
tff(fact_1082_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
      <=> pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ) ).

% both_member_options_equiv_member
tff(fact_1083_Leaf__0__not,axiom,
    ! [A2: bool,B3: bool] : ~ pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Leaf(A2,B3)),zero_zero_nat)) ).

% Leaf_0_not
tff(fact_1084_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_delete(T,X)),Y))
      <=> ( ( X != Y )
          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),Y)) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_1085_maxbmo,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_maxt(T) = some_nat(X) )
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X)) ) ).

% maxbmo
tff(fact_1086_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_1087_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_1088_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_1089_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_Pr2477239776880934807at_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(X,Xa2,Xb) = Y )
     => ( ( ( Xa2 = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] : Xa2 = some_P7363390416028606310at_nat(V2)
           => ( ( Xb = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A: product_prod_nat_nat] :
                ( ( Xa2 = some_P7363390416028606310at_nat(A) )
               => ! [B: product_prod_nat_nat] :
                    ( ( Xb = some_P7363390416028606310at_nat(B) )
                   => ( Y != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(X,A),B)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1090_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_num_fun_num_num,Xa2: option_num,Xb: option_num,Y: option_num] :
      ( ( vEBT_V819420779217536731ft_num(X,Xa2,Xb) = Y )
     => ( ( ( Xa2 = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V2: num] : Xa2 = some_num(V2)
           => ( ( Xb = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A: num] :
                ( ( Xa2 = some_num(A) )
               => ! [B: num] :
                    ( ( Xb = some_num(B) )
                   => ( Y != some_num(aa_num_num(aa_num_fun_num_num(X,A),B)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1091_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa2: option_nat,Xb: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(X),Xa2),Xb) = Y )
     => ( ( ( Xa2 = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] : Xa2 = some_nat(V2)
           => ( ( Xb = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A: nat] :
                ( ( Xa2 = some_nat(A) )
               => ! [B: nat] :
                    ( ( Xb = some_nat(B) )
                   => ( Y != some_nat(aa_nat_nat(aa_nat_fun_nat_nat(X,A),B)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1092_Set_Ois__empty__def,axiom,
    ! [A3: set_real] :
      ( pp(aa_set_real_bool(is_empty_real,A3))
    <=> ( A3 = bot_bot_set_real ) ) ).

% Set.is_empty_def
tff(fact_1093_Set_Ois__empty__def,axiom,
    ! [A3: set_bool] :
      ( pp(aa_set_bool_bool(is_empty_bool,A3))
    <=> ( A3 = bot_bot_set_bool ) ) ).

% Set.is_empty_def
tff(fact_1094_Set_Ois__empty__def,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(is_empty_nat,A3))
    <=> ( A3 = bot_bot_set_nat ) ) ).

% Set.is_empty_def
tff(fact_1095_Set_Ois__empty__def,axiom,
    ! [A3: set_int] :
      ( pp(aa_set_int_bool(is_empty_int,A3))
    <=> ( A3 = bot_bot_set_int ) ) ).

% Set.is_empty_def
tff(fact_1096_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_1097_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_1098_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_1099_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_1100_enumerate__mono__iff,axiom,
    ! [S: set_Extended_enat,M: nat,N: nat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,infini7641415182203889163d_enat(S,M)),infini7641415182203889163d_enat(S,N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% enumerate_mono_iff
tff(fact_1101_enumerate__mono__iff,axiom,
    ! [S: set_nat,M: nat,N: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S),M)),aa_nat_nat(infini8530281810654367211te_nat(S),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% enumerate_mono_iff
tff(fact_1102_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_12)) ) ).

% not_min_Null_member
tff(fact_1103_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_1104_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_1105_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_1106_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_1107_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_1108_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_1109_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_1110_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_1111_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_1112_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_1113_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_1114_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_1115_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_1116_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_1117_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_1118_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_1119_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_1120_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_1121_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_1122_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_1123_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_1124_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_1125_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1126_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1127_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1128_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1129_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_1130_nat__int__comparison_I2_J,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B3))) ) ).

% nat_int_comparison(2)
tff(fact_1131_nat__int__comparison_I3_J,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B3))) ) ).

% nat_int_comparison(3)
tff(fact_1132_enumerate__in__set,axiom,
    ! [S: set_Extended_enat,N: nat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => pp(aa_set3347977610044113549t_bool(member_Extended_enat(infini7641415182203889163d_enat(S,N)),S)) ) ).

% enumerate_in_set
tff(fact_1133_enumerate__in__set,axiom,
    ! [S: set_nat,N: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
     => pp(aa_set_nat_bool(member_nat(aa_nat_nat(infini8530281810654367211te_nat(S),N)),S)) ) ).

% enumerate_in_set
tff(fact_1134_enumerate__Ex,axiom,
    ! [S: set_nat,S2: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_nat_bool(member_nat(S2),S))
       => ? [N2: nat] : aa_nat_nat(infini8530281810654367211te_nat(S),N2) = S2 ) ) ).

% enumerate_Ex
tff(fact_1135_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% of_nat_0_le_iff
tff(fact_1136_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N))) ).

% of_nat_0_le_iff
tff(fact_1137_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).

% of_nat_0_le_iff
tff(fact_1138_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% of_nat_0_le_iff
tff(fact_1139_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)) ).

% of_nat_less_0_iff
tff(fact_1140_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)) ).

% of_nat_less_0_iff
tff(fact_1141_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)) ).

% of_nat_less_0_iff
tff(fact_1142_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real)) ).

% of_nat_less_0_iff
tff(fact_1143_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_1144_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_1145_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_1146_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_1147_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))) ) ).

% less_imp_of_nat_less
tff(fact_1148_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) ) ).

% less_imp_of_nat_less
tff(fact_1149_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% less_imp_of_nat_less
tff(fact_1150_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N))) ) ).

% less_imp_of_nat_less
tff(fact_1151_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J))) ) ).

% of_nat_mono
tff(fact_1152_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J))) ) ).

% of_nat_mono
tff(fact_1153_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J))) ) ).

% of_nat_mono
tff(fact_1154_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J))) ) ).

% of_nat_mono
tff(fact_1155_le__enumerate,axiom,
    ! [S: set_nat,N: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(infini8530281810654367211te_nat(S),N))) ) ).

% le_enumerate
tff(fact_1156_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_Pr2477239776880934807at_nat,A2: product_prod_nat_nat,B3: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(F,some_P7363390416028606310at_nat(A2),some_P7363390416028606310at_nat(B3)) = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(F,A2),B3)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1157_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_num_fun_num_num,A2: num,B3: num] : vEBT_V819420779217536731ft_num(F,some_num(A2),some_num(B3)) = some_num(aa_num_num(aa_num_fun_num_num(F,A2),B3)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1158_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_nat_fun_nat_nat,A2: nat,B3: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(F),some_nat(A2)),some_nat(B3)) = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(F,A2),B3)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1159_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_1160_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_1161_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_1162_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool] : vEBT_vebt_pred(vEBT_Leaf(Uu,Uv),zero_zero_nat) = none_nat ).

% vebt_pred.simps(1)
tff(fact_1163_enumerate__mono,axiom,
    ! [M: nat,N: nat,S: set_Extended_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,infini7641415182203889163d_enat(S,M)),infini7641415182203889163d_enat(S,N))) ) ) ).

% enumerate_mono
tff(fact_1164_enumerate__mono,axiom,
    ! [M: nat,N: nat,S: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S),M)),aa_nat_nat(infini8530281810654367211te_nat(S),N))) ) ) ).

% enumerate_mono
tff(fact_1165_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(1)
tff(fact_1166_vebt__delete_Osimps_I1_J,axiom,
    ! [A2: bool,B3: bool] : vEBT_vebt_delete(vEBT_Leaf(A2,B3),zero_zero_nat) = vEBT_Leaf(fFalse,B3) ).

% vebt_delete.simps(1)
tff(fact_1167_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,Uw: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Leaf(Uu,Uv)),Uw)) ).

% VEBT_internal.membermima.simps(1)
tff(fact_1168_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(Uu,fTrue))) ).

% VEBT_internal.minNull.simps(3)
tff(fact_1169_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(fTrue,Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_1170_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(fFalse,fFalse))) ).

% VEBT_internal.minNull.simps(1)
tff(fact_1171_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
    <=> ? [A6: bool,B6: bool] : T = vEBT_Leaf(A6,B6) ) ).

% deg1Leaf
tff(fact_1172_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
     => ? [A: bool,B: bool] : T = vEBT_Leaf(A,B) ) ).

% deg_1_Leaf
tff(fact_1173_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( N = one_one_nat )
       => ? [A: bool,B: bool] : T = vEBT_Leaf(A,B) ) ) ).

% deg_1_Leafy
tff(fact_1174_pos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ~ ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% pos_int_cases
tff(fact_1175_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_1176_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_1177_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_1178_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_1179_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_1180_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_1181_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_1182_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_1183_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_1184_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_1185_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_1186_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,N) = one_one_complex )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_1187_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,N) = one_one_rat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_1188_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,N) = one_one_nat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_1189_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,N) = one_one_int )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_1190_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,N) = one_one_real )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_1191_less__one,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),one_one_nat))
    <=> ( N = zero_zero_nat ) ) ).

% less_one
tff(fact_1192_nonneg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ~ ! [N2: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% nonneg_int_cases
tff(fact_1193_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ? [N2: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% zero_le_imp_eq_int
tff(fact_1194_less__eq__int__code_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

% less_eq_int_code(1)
tff(fact_1195_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
    <=> ( X = one_one_complex ) ) ).

% one_reorient
tff(fact_1196_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
    <=> ( X = one_one_real ) ) ).

% one_reorient
tff(fact_1197_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
    <=> ( X = one_one_rat ) ) ).

% one_reorient
tff(fact_1198_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_1199_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_1200_less__int__code_I1_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).

% less_int_code(1)
tff(fact_1201_verit__la__generic,axiom,
    ! [A2: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),X))
      | ( A2 = X )
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A2)) ) ).

% verit_la_generic
tff(fact_1202_imp__le__cong,axiom,
    ! [X: int,X7: int,P: bool,P2: bool] :
      ( ( X = X7 )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X7))
         => ( pp(P)
          <=> pp(P2) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
           => pp(P) )
        <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X7))
           => pp(P2) ) ) ) ) ).

% imp_le_cong
tff(fact_1203_conj__le__cong,axiom,
    ! [X: int,X7: int,P: bool,P2: bool] :
      ( ( X = X7 )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X7))
         => ( pp(P)
          <=> pp(P2) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
            & pp(P) )
        <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X7))
            & pp(P2) ) ) ) ) ).

% conj_le_cong
tff(fact_1204_le__numeral__extra_I4_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),one_one_real)) ).

% le_numeral_extra(4)
tff(fact_1205_le__numeral__extra_I4_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),one_one_rat)) ).

% le_numeral_extra(4)
tff(fact_1206_le__numeral__extra_I4_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),one_one_nat)) ).

% le_numeral_extra(4)
tff(fact_1207_le__numeral__extra_I4_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),one_one_int)) ).

% le_numeral_extra(4)
tff(fact_1208_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),one_one_real)) ).

% less_numeral_extra(4)
tff(fact_1209_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),one_one_rat)) ).

% less_numeral_extra(4)
tff(fact_1210_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),one_one_nat)) ).

% less_numeral_extra(4)
tff(fact_1211_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),one_one_int)) ).

% less_numeral_extra(4)
tff(fact_1212_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Leaf(Uu,Uv)),D))
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_1213_less__numeral__extra_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

% less_numeral_extra(1)
tff(fact_1214_less__numeral__extra_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).

% less_numeral_extra(1)
tff(fact_1215_less__numeral__extra_I1_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% less_numeral_extra(1)
tff(fact_1216_less__numeral__extra_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% less_numeral_extra(1)
tff(fact_1217_vebt__member_Osimps_I1_J,axiom,
    ! [A2: bool,B3: bool,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Leaf(A2,B3)),X))
    <=> ( ( ( X = zero_zero_nat )
         => pp(A2) )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => pp(B3) )
            & ( X = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
tff(fact_1218_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: bool,B3: bool,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Leaf(A2,B3)),X))
    <=> ( ( ( X = zero_zero_nat )
         => pp(A2) )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => pp(B3) )
            & ( X = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_1219_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% zle_int
tff(fact_1220_vebt__mint_Osimps_I1_J,axiom,
    ! [A2: bool,B3: bool] :
      ( ( pp(A2)
       => ( vEBT_vebt_mint(vEBT_Leaf(A2,B3)) = some_nat(zero_zero_nat) ) )
      & ( ~ pp(A2)
       => ( ( pp(B3)
           => ( vEBT_vebt_mint(vEBT_Leaf(A2,B3)) = some_nat(one_one_nat) ) )
          & ( ~ pp(B3)
           => ( vEBT_vebt_mint(vEBT_Leaf(A2,B3)) = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
tff(fact_1221_vebt__maxt_Osimps_I1_J,axiom,
    ! [B3: bool,A2: bool] :
      ( ( pp(B3)
       => ( vEBT_vebt_maxt(vEBT_Leaf(A2,B3)) = some_nat(one_one_nat) ) )
      & ( ~ pp(B3)
       => ( ( pp(A2)
           => ( vEBT_vebt_maxt(vEBT_Leaf(A2,B3)) = some_nat(zero_zero_nat) ) )
          & ( ~ pp(A2)
           => ( vEBT_vebt_maxt(vEBT_Leaf(A2,B3)) = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
tff(fact_1222_vebt__succ_Osimps_I1_J,axiom,
    ! [B3: bool,Uu: bool] :
      ( ( pp(B3)
       => ( vEBT_vebt_succ(vEBT_Leaf(Uu,B3),zero_zero_nat) = some_nat(one_one_nat) ) )
      & ( ~ pp(B3)
       => ( vEBT_vebt_succ(vEBT_Leaf(Uu,B3),zero_zero_nat) = none_nat ) ) ) ).

% vebt_succ.simps(1)
tff(fact_1223_not__one__less__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),zero_zero_real)) ).

% not_one_less_zero
tff(fact_1224_not__one__less__zero,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),zero_zero_rat)) ).

% not_one_less_zero
tff(fact_1225_not__one__less__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).

% not_one_less_zero
tff(fact_1226_not__one__less__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),zero_zero_int)) ).

% not_one_less_zero
tff(fact_1227_zero__less__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

% zero_less_one
tff(fact_1228_zero__less__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).

% zero_less_one
tff(fact_1229_zero__less__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one
tff(fact_1230_zero__less__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% zero_less_one
tff(fact_1231_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

% zero_less_one_class.zero_le_one
tff(fact_1232_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).

% zero_less_one_class.zero_le_one
tff(fact_1233_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one_class.zero_le_one
tff(fact_1234_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% zero_less_one_class.zero_le_one
tff(fact_1235_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1236_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1237_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1238_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1239_not__one__le__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),zero_zero_real)) ).

% not_one_le_zero
tff(fact_1240_not__one__le__zero,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),zero_zero_rat)) ).

% not_one_le_zero
tff(fact_1241_not__one__le__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).

% not_one_le_zero
tff(fact_1242_not__one__le__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).

% not_one_le_zero
tff(fact_1243_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).

% reals_Archimedean2
tff(fact_1244_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,N2))) ).

% reals_Archimedean2
tff(fact_1245_real__arch__simple,axiom,
    ! [X: real] :
    ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N2))) ).

% real_arch_simple
tff(fact_1246_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).

% real_arch_simple
tff(fact_1247_vebt__pred_Osimps_I3_J,axiom,
    ! [B3: bool,A2: bool,Va: nat] :
      ( ( pp(B3)
       => ( vEBT_vebt_pred(vEBT_Leaf(A2,B3),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = some_nat(one_one_nat) ) )
      & ( ~ pp(B3)
       => ( ( pp(A2)
           => ( vEBT_vebt_pred(vEBT_Leaf(A2,B3),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = some_nat(zero_zero_nat) ) )
          & ( ~ pp(A2)
           => ( vEBT_vebt_pred(vEBT_Leaf(A2,B3),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
tff(fact_1248_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_1249_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_1250_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_1251_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_1252_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_1253_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_1254_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_1255_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_1256_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_1257_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: bool,X222: bool] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf(X21,X222)) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_1258_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_1259_nat_Oinject,axiom,
    ! [X2: nat,Y22: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y22) )
    <=> ( X2 = Y22 ) ) ).

% nat.inject
tff(fact_1260_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_less_eq
tff(fact_1261_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) ) ).

% Suc_mono
tff(fact_1262_lessI,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,N))) ).

% lessI
tff(fact_1263_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).

% Suc_le_mono
tff(fact_1264_less__Suc0,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( N = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_1265_zero__less__Suc,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,N))) ).

% zero_less_Suc
tff(fact_1266_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa_nat_nat(suc,N) ).

% n_not_Suc_n
tff(fact_1267_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_1268_exists__least__lemma,axiom,
    ! [P: fun_nat_bool] :
      ( ~ pp(aa_nat_bool(P,zero_zero_nat))
     => ( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
       => ? [N2: nat] :
            ( ~ pp(aa_nat_bool(P,N2))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) ) ).

% exists_least_lemma
tff(fact_1269_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).

% nat.distinct(1)
tff(fact_1270_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_1271_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_1272_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa_nat_nat(suc,X2) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_1273_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_1274_nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,N2))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% nat_induct
tff(fact_1275_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [X5: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X5),zero_zero_nat))
     => ( ! [Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,zero_zero_nat),aa_nat_nat(suc,Y3)))
       => ( ! [X5: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X5),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(suc,X5)),aa_nat_nat(suc,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ) ).

% diff_induct
tff(fact_1276_zero__induct,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(P,N2)) )
       => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ).

% zero_induct
tff(fact_1277_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_1278_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_1279_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_1280_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M4: nat] : N = aa_nat_nat(suc,M4) ) ).

% not0_implies_Suc
tff(fact_1281_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va2: nat] : X != aa_nat_nat(suc,aa_nat_nat(suc,Va2)) ) ) ).

% vebt_buildup.cases
tff(fact_1282_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
      <=> ( N = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_1283_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I2: nat] :
            ( ( J = aa_nat_nat(suc,I2) )
           => pp(aa_nat_bool(P,I2)) )
       => ( ! [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
             => ( pp(aa_nat_bool(P,aa_nat_nat(suc,I2)))
               => pp(aa_nat_bool(P,I2)) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% strict_inc_induct
tff(fact_1284_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),aa_nat_nat(suc,I2)))
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),K2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),J2))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,J2),K2))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),K2)) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I),J)) ) ) ) ).

% less_Suc_induct
tff(fact_1285_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K)) ) ) ).

% less_trans_Suc
tff(fact_1286_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_less_SucD
tff(fact_1287_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
       => ( M = N ) ) ) ).

% less_antisym
tff(fact_1288_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
    <=> ? [M6: nat] :
          ( ( M = aa_nat_nat(suc,M6) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M6)) ) ) ).

% Suc_less_eq2
tff(fact_1289_All__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,N))
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
           => pp(aa_nat_bool(P,I4)) ) ) ) ).

% All_less_Suc
tff(fact_1290_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M))) ) ).

% not_less_eq
tff(fact_1291_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) ) ) ).

% less_Suc_eq
tff(fact_1292_Ex__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,N))
        | ? [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
            & pp(aa_nat_bool(P,I4)) ) ) ) ).

% Ex_less_Suc
tff(fact_1293_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).

% less_SucI
tff(fact_1294_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( M = N ) ) ) ).

% less_SucE
tff(fact_1295_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( ( aa_nat_nat(suc,M) != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N)) ) ) ).

% Suc_lessI
tff(fact_1296_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K))
     => ~ ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_1297_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_lessD
tff(fact_1298_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K))
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_1299_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% Suc_leD
tff(fact_1300_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_SucE
tff(fact_1301_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N))) ) ).

% le_SucI
tff(fact_1302_Suc__le__D,axiom,
    ! [N: nat,M7: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M7))
     => ? [M4: nat] : M7 = aa_nat_nat(suc,M4) ) ).

% Suc_le_D
tff(fact_1303_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        | ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_1304_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),N)) ).

% Suc_n_not_le_n
tff(fact_1305_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).

% not_less_eq_eq
tff(fact_1306_full__nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N2))
             => pp(aa_nat_bool(P,M2)) )
         => pp(aa_nat_bool(P,N2)) )
     => pp(aa_nat_bool(P,N)) ) ).

% full_nat_induct
tff(fact_1307_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(P,M))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
             => ( pp(aa_nat_bool(P,N2))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_1308_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( ! [X5: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X5),X5))
       => ( ! [X5: nat,Y3: nat,Z3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X5),Y3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y3),Z3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X5),Z3)) ) )
         => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,N2),aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,M),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_1309_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1310_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1311_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1312_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1313_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1314_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_1315_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_1316_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_1317_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_1318_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_1319_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_int,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N2)),aa_nat_set_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N)),aa_nat_set_int(F,N6))) ) ) ).

% lift_Suc_mono_le
tff(fact_1320_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N6))) ) ) ).

% lift_Suc_mono_le
tff(fact_1321_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,N6))) ) ) ).

% lift_Suc_mono_le
tff(fact_1322_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N6))) ) ) ).

% lift_Suc_mono_le
tff(fact_1323_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,N6))) ) ) ).

% lift_Suc_mono_le
tff(fact_1324_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_int,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,aa_nat_nat(suc,N2))),aa_nat_set_int(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N6)),aa_nat_set_int(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1325_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N2))),aa_nat_rat(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N6)),aa_nat_rat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1326_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N2))),aa_nat_num(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N6)),aa_nat_num(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1327_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N2))),aa_nat_nat(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N6)),aa_nat_nat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1328_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N2))),aa_nat_int(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N6)),aa_nat_int(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1329_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_1330_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_1331_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_1332_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_1333_Ex__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        | ? [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,I4))) ) ) ) ).

% Ex_less_Suc2
tff(fact_1334_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ? [M3: nat] : N = aa_nat_nat(suc,M3) ) ).

% gr0_conv_Suc
tff(fact_1335_All__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,I4))) ) ) ) ).

% All_less_Suc2
tff(fact_1336_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ? [M4: nat] : N = aa_nat_nat(suc,M4) ) ).

% gr0_implies_Suc
tff(fact_1337_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M = aa_nat_nat(suc,J3) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_1338_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N)) ) ).

% Suc_leI
tff(fact_1339_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_le_eq
tff(fact_1340_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(P,I))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J))
               => ( pp(aa_nat_bool(P,N2))
                 => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) )
         => pp(aa_nat_bool(P,J)) ) ) ) ).

% dec_induct
tff(fact_1341_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(P,J))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J))
               => ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
                 => pp(aa_nat_bool(P,N2)) ) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% inc_induct
tff(fact_1342_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_le_lessD
tff(fact_1343_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
      <=> ( N = M ) ) ) ).

% le_less_Suc_eq
tff(fact_1344_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_Suc_eq_le
tff(fact_1345_less__eq__Suc__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).

% less_eq_Suc_le
tff(fact_1346_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).

% le_imp_less_Suc
tff(fact_1347_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_1348_vebt__delete_Osimps_I3_J,axiom,
    ! [A2: bool,B3: bool,N: nat] : vEBT_vebt_delete(vEBT_Leaf(A2,B3),aa_nat_nat(suc,aa_nat_nat(suc,N))) = vEBT_Leaf(A2,B3) ).

% vebt_delete.simps(3)
tff(fact_1349_ex__least__nat__less,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N))
            & ! [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),K2))
               => ~ pp(aa_nat_bool(P,I3)) )
            & pp(aa_nat_bool(P,aa_nat_nat(suc,K2))) ) ) ) ).

% ex_least_nat_less
tff(fact_1350_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1351_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1352_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1353_nat__induct__non__zero,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(P,one_one_nat))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => ( pp(aa_nat_bool(P,N2))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_1354_enumerate__step,axiom,
    ! [S: set_Extended_enat,N: nat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,infini7641415182203889163d_enat(S,N)),infini7641415182203889163d_enat(S,aa_nat_nat(suc,N)))) ) ).

% enumerate_step
tff(fact_1355_enumerate__step,axiom,
    ! [S: set_nat,N: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S),N)),aa_nat_nat(infini8530281810654367211te_nat(S),aa_nat_nat(suc,N)))) ) ).

% enumerate_step
tff(fact_1356_invar__vebt_Ointros_I1_J,axiom,
    ! [A2: bool,B3: bool] : pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Leaf(A2,B3)),aa_nat_nat(suc,zero_zero_nat))) ).

% invar_vebt.intros(1)
tff(fact_1357_vebt__delete_Osimps_I2_J,axiom,
    ! [A2: bool,B3: bool] : vEBT_vebt_delete(vEBT_Leaf(A2,B3),aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf(A2,fFalse) ).

% vebt_delete.simps(2)
tff(fact_1358_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% int_one_le_iff_zero_less
tff(fact_1359_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(2)
tff(fact_1360_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: bool,Uw: bool,N: nat] : vEBT_vebt_succ(vEBT_Leaf(Uv,Uw),aa_nat_nat(suc,N)) = none_nat ).

% vebt_succ.simps(2)
tff(fact_1361_vebt__pred_Osimps_I2_J,axiom,
    ! [A2: bool,Uw: bool] :
      ( ( pp(A2)
       => ( vEBT_vebt_pred(vEBT_Leaf(A2,Uw),aa_nat_nat(suc,zero_zero_nat)) = some_nat(zero_zero_nat) ) )
      & ( ~ pp(A2)
       => ( vEBT_vebt_pred(vEBT_Leaf(A2,Uw),aa_nat_nat(suc,zero_zero_nat)) = none_nat ) ) ) ).

% vebt_pred.simps(2)
tff(fact_1362_option_Osize__gen_I1_J,axiom,
    ! [X: fun_nat_nat] : aa_option_nat_nat(size_option_nat(X),none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_1363_option_Osize__gen_I1_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(X,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_1364_option_Osize__gen_I1_J,axiom,
    ! [X: fun_num_nat] : size_option_num(X,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_1365_option_Osize_I3_J,axiom,
    aa_option_nat_nat(size_size_option_nat,none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1366_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1367_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1368_option_Osize_I4_J,axiom,
    ! [X2: nat] : aa_option_nat_nat(size_size_option_nat,some_nat(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_1369_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_1370_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_1371_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N2: nat] : X != aa_nat_nat(suc,N2) ) ).

% list_decode.cases
tff(fact_1372_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,N))))
     => ? [Info: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] : Tree = vEBT_Node(Info,aa_nat_nat(suc,aa_nat_nat(suc,N)),TreeList,S3) ) ).

% deg_SUcn_Node
tff(fact_1373_neg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ~ ! [N2: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% neg_int_cases
tff(fact_1374_frac__eq,axiom,
    ! [X: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = X )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% frac_eq
tff(fact_1375_frac__eq,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = X )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ) ).

% frac_eq
tff(fact_1376_finite__enum__subset,axiom,
    ! [X6: set_Extended_enat,Y5: set_Extended_enat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),finite121521170596916366d_enat(X6)))
         => ( infini7641415182203889163d_enat(X6,I2) = infini7641415182203889163d_enat(Y5,I2) ) )
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,X6))
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,Y5))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite121521170596916366d_enat(X6)),finite121521170596916366d_enat(Y5)))
           => pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(X6),Y5)) ) ) ) ) ).

% finite_enum_subset
tff(fact_1377_finite__enum__subset,axiom,
    ! [X6: set_nat,Y5: set_nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_set_nat_nat(finite_card_nat,X6)))
         => ( aa_nat_nat(infini8530281810654367211te_nat(X6),I2) = aa_nat_nat(infini8530281810654367211te_nat(Y5),I2) ) )
     => ( pp(aa_set_nat_bool(finite_finite_nat,X6))
       => ( pp(aa_set_nat_bool(finite_finite_nat,Y5))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,X6)),aa_set_nat_nat(finite_card_nat,Y5)))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X6),Y5)) ) ) ) ) ).

% finite_enum_subset
tff(fact_1378_Suc__diff__1,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N ) ) ).

% Suc_diff_1
tff(fact_1379_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( N != zero_zero_nat )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N)))) ) ) ).

% inverse_of_nat_le
tff(fact_1380_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( N != zero_zero_nat )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,N)))) ) ) ).

% inverse_of_nat_le
tff(fact_1381_deg__deg__n,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary)),N))
     => ( Deg = N ) ) ).

% deg_deg_n
tff(fact_1382_add_Oinverse__inverse,axiom,
    ! [A2: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,A2)) = A2 ).

% add.inverse_inverse
tff(fact_1383_add_Oinverse__inverse,axiom,
    ! [A2: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,A2)) = A2 ).

% add.inverse_inverse
tff(fact_1384_add_Oinverse__inverse,axiom,
    ! [A2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,A2)) = A2 ).

% add.inverse_inverse
tff(fact_1385_add_Oinverse__inverse,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)) = A2 ).

% add.inverse_inverse
tff(fact_1386_add_Oinverse__inverse,axiom,
    ! [A2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A2)) = A2 ).

% add.inverse_inverse
tff(fact_1387_neg__equal__iff__equal,axiom,
    ! [A2: int,B3: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = aa_int_int(uminus_uminus_int,B3) )
    <=> ( A2 = B3 ) ) ).

% neg_equal_iff_equal
tff(fact_1388_neg__equal__iff__equal,axiom,
    ! [A2: real,B3: real] :
      ( ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(uminus_uminus_real,B3) )
    <=> ( A2 = B3 ) ) ).

% neg_equal_iff_equal
tff(fact_1389_neg__equal__iff__equal,axiom,
    ! [A2: rat,B3: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A2) = aa_rat_rat(uminus_uminus_rat,B3) )
    <=> ( A2 = B3 ) ) ).

% neg_equal_iff_equal
tff(fact_1390_neg__equal__iff__equal,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3) )
    <=> ( A2 = B3 ) ) ).

% neg_equal_iff_equal
tff(fact_1391_neg__equal__iff__equal,axiom,
    ! [A2: complex,B3: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A2) = aa_complex_complex(uminus1482373934393186551omplex,B3) )
    <=> ( A2 = B3 ) ) ).

% neg_equal_iff_equal
tff(fact_1392_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),A2) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1393_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),A2) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1394_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),A2) = zero_zero_nat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1395_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),A2) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1396_diff__zero,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),zero_zero_real) = A2 ).

% diff_zero
tff(fact_1397_diff__zero,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),zero_zero_rat) = A2 ).

% diff_zero
tff(fact_1398_diff__zero,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),zero_zero_nat) = A2 ).

% diff_zero
tff(fact_1399_diff__zero,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),zero_zero_int) = A2 ).

% diff_zero
tff(fact_1400_zero__diff,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),A2) = zero_zero_nat ).

% zero_diff
tff(fact_1401_diff__0__right,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),zero_zero_real) = A2 ).

% diff_0_right
tff(fact_1402_diff__0__right,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),zero_zero_rat) = A2 ).

% diff_0_right
tff(fact_1403_diff__0__right,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),zero_zero_int) = A2 ).

% diff_0_right
tff(fact_1404_diff__self,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),A2) = zero_zero_real ).

% diff_self
tff(fact_1405_diff__self,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),A2) = zero_zero_rat ).

% diff_self
tff(fact_1406_diff__self,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),A2) = zero_zero_int ).

% diff_self
tff(fact_1407_div__by__0,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_1408_div__by__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_1409_div__by__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),zero_zero_nat) = zero_zero_nat ).

% div_by_0
tff(fact_1410_div__by__0,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_1411_div__by__0,axiom,
    ! [A2: complex] : aa_complex_complex(divide1717551699836669952omplex(A2),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_1412_div__0,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,zero_zero_rat),A2) = zero_zero_rat ).

% div_0
tff(fact_1413_div__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A2) = zero_zero_int ).

% div_0
tff(fact_1414_div__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A2) = zero_zero_nat ).

% div_0
tff(fact_1415_div__0,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,zero_zero_real),A2) = zero_zero_real ).

% div_0
tff(fact_1416_div__0,axiom,
    ! [A2: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A2) = zero_zero_complex ).

% div_0
tff(fact_1417_neg__le__iff__le,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B3)),aa_real_real(uminus_uminus_real,A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ).

% neg_le_iff_le
tff(fact_1418_neg__le__iff__le,axiom,
    ! [B3: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B3)) ) ).

% neg_le_iff_le
tff(fact_1419_neg__le__iff__le,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B3)),aa_rat_rat(uminus_uminus_rat,A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ).

% neg_le_iff_le
tff(fact_1420_neg__le__iff__le,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B3)),aa_int_int(uminus_uminus_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ).

% neg_le_iff_le
tff(fact_1421_neg__equal__zero,axiom,
    ! [A2: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = A2 )
    <=> ( A2 = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_1422_neg__equal__zero,axiom,
    ! [A2: real] :
      ( ( aa_real_real(uminus_uminus_real,A2) = A2 )
    <=> ( A2 = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_1423_neg__equal__zero,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A2) = A2 )
    <=> ( A2 = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_1424_neg__equal__zero,axiom,
    ! [A2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = A2 )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_1425_equal__neg__zero,axiom,
    ! [A2: int] :
      ( ( A2 = aa_int_int(uminus_uminus_int,A2) )
    <=> ( A2 = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_1426_equal__neg__zero,axiom,
    ! [A2: real] :
      ( ( A2 = aa_real_real(uminus_uminus_real,A2) )
    <=> ( A2 = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_1427_equal__neg__zero,axiom,
    ! [A2: rat] :
      ( ( A2 = aa_rat_rat(uminus_uminus_rat,A2) )
    <=> ( A2 = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_1428_equal__neg__zero,axiom,
    ! [A2: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_1429_neg__equal__0__iff__equal,axiom,
    ! [A2: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
tff(fact_1430_neg__equal__0__iff__equal,axiom,
    ! [A2: real] :
      ( ( aa_real_real(uminus_uminus_real,A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
tff(fact_1431_neg__equal__0__iff__equal,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_1432_neg__equal__0__iff__equal,axiom,
    ! [A2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = zero_z3403309356797280102nteger )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_1433_neg__equal__0__iff__equal,axiom,
    ! [A2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A2) = zero_zero_complex )
    <=> ( A2 = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
tff(fact_1434_neg__0__equal__iff__equal,axiom,
    ! [A2: int] :
      ( ( zero_zero_int = aa_int_int(uminus_uminus_int,A2) )
    <=> ( zero_zero_int = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_1435_neg__0__equal__iff__equal,axiom,
    ! [A2: real] :
      ( ( zero_zero_real = aa_real_real(uminus_uminus_real,A2) )
    <=> ( zero_zero_real = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_1436_neg__0__equal__iff__equal,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat = aa_rat_rat(uminus_uminus_rat,A2) )
    <=> ( zero_zero_rat = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_1437_neg__0__equal__iff__equal,axiom,
    ! [A2: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) )
    <=> ( zero_z3403309356797280102nteger = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_1438_neg__0__equal__iff__equal,axiom,
    ! [A2: complex] :
      ( ( zero_zero_complex = aa_complex_complex(uminus1482373934393186551omplex,A2) )
    <=> ( zero_zero_complex = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_1439_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_1440_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_1441_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_1442_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_1443_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_1444_neg__less__iff__less,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B3)),aa_int_int(uminus_uminus_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3)) ) ).

% neg_less_iff_less
tff(fact_1445_neg__less__iff__less,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B3)),aa_real_real(uminus_uminus_real,A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) ) ).

% neg_less_iff_less
tff(fact_1446_neg__less__iff__less,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B3)),aa_rat_rat(uminus_uminus_rat,A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) ) ).

% neg_less_iff_less
tff(fact_1447_neg__less__iff__less,axiom,
    ! [B3: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B3)) ) ).

% neg_less_iff_less
tff(fact_1448_minus__diff__eq,axiom,
    ! [A2: int,B3: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B3),A2) ).

% minus_diff_eq
tff(fact_1449_minus__diff__eq,axiom,
    ! [A2: real,B3: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),A2) ).

% minus_diff_eq
tff(fact_1450_minus__diff__eq,axiom,
    ! [A2: rat,B3: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B3),A2) ).

% minus_diff_eq
tff(fact_1451_minus__diff__eq,axiom,
    ! [A2: code_integer,B3: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A2,B3)) = minus_8373710615458151222nteger(B3,A2) ).

% minus_diff_eq
tff(fact_1452_minus__diff__eq,axiom,
    ! [A2: complex,B3: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A2),B3)) = aa_complex_complex(minus_minus_complex(B3),A2) ).

% minus_diff_eq
tff(fact_1453_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% diff_Suc_Suc
tff(fact_1454_Suc__diff__diff,axiom,
    ! [M: nat,N: 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)),N)),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),N)),K) ).

% Suc_diff_diff
tff(fact_1455_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_1456_diff__0__eq__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_1457_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_1458_diff__ge__0__iff__ge,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2)) ) ).

% diff_ge_0_iff_ge
tff(fact_1459_diff__ge__0__iff__ge,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2)) ) ).

% diff_ge_0_iff_ge
tff(fact_1460_diff__ge__0__iff__ge,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2)) ) ).

% diff_ge_0_iff_ge
tff(fact_1461_diff__gt__0__iff__gt,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2)) ) ).

% diff_gt_0_iff_gt
tff(fact_1462_diff__gt__0__iff__gt,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2)) ) ).

% diff_gt_0_iff_gt
tff(fact_1463_diff__gt__0__iff__gt,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2)) ) ).

% diff_gt_0_iff_gt
tff(fact_1464_neg__0__le__iff__le,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) ) ).

% neg_0_le_iff_le
tff(fact_1465_neg__0__le__iff__le,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger)) ) ).

% neg_0_le_iff_le
tff(fact_1466_neg__0__le__iff__le,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) ) ).

% neg_0_le_iff_le
tff(fact_1467_neg__0__le__iff__le,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) ) ).

% neg_0_le_iff_le
tff(fact_1468_neg__le__0__iff__le,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2)) ) ).

% neg_le_0_iff_le
tff(fact_1469_neg__le__0__iff__le,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)) ) ).

% neg_le_0_iff_le
tff(fact_1470_neg__le__0__iff__le,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2)) ) ).

% neg_le_0_iff_le
tff(fact_1471_neg__le__0__iff__le,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2)) ) ).

% neg_le_0_iff_le
tff(fact_1472_less__eq__neg__nonpos,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(uminus_uminus_real,A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) ) ).

% less_eq_neg_nonpos
tff(fact_1473_less__eq__neg__nonpos,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger)) ) ).

% less_eq_neg_nonpos
tff(fact_1474_less__eq__neg__nonpos,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(uminus_uminus_rat,A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) ) ).

% less_eq_neg_nonpos
tff(fact_1475_less__eq__neg__nonpos,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(uminus_uminus_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) ) ).

% less_eq_neg_nonpos
tff(fact_1476_neg__less__eq__nonneg,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),A2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2)) ) ).

% neg_less_eq_nonneg
tff(fact_1477_neg__less__eq__nonneg,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),A2))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)) ) ).

% neg_less_eq_nonneg
tff(fact_1478_neg__less__eq__nonneg,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),A2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2)) ) ).

% neg_less_eq_nonneg
tff(fact_1479_neg__less__eq__nonneg,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),A2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2)) ) ).

% neg_less_eq_nonneg
tff(fact_1480_neg__less__0__iff__less,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2)) ) ).

% neg_less_0_iff_less
tff(fact_1481_neg__less__0__iff__less,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2)) ) ).

% neg_less_0_iff_less
tff(fact_1482_neg__less__0__iff__less,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A2)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2)) ) ).

% neg_less_0_iff_less
tff(fact_1483_neg__less__0__iff__less,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)) ) ).

% neg_less_0_iff_less
tff(fact_1484_neg__0__less__iff__less,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ).

% neg_0_less_iff_less
tff(fact_1485_neg__0__less__iff__less,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ).

% neg_0_less_iff_less
tff(fact_1486_neg__0__less__iff__less,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ).

% neg_0_less_iff_less
tff(fact_1487_neg__0__less__iff__less,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger)) ) ).

% neg_0_less_iff_less
tff(fact_1488_neg__less__pos,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A2)),A2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2)) ) ).

% neg_less_pos
tff(fact_1489_neg__less__pos,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A2)),A2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2)) ) ).

% neg_less_pos
tff(fact_1490_neg__less__pos,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A2)),A2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2)) ) ).

% neg_less_pos
tff(fact_1491_neg__less__pos,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),A2))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)) ) ).

% neg_less_pos
tff(fact_1492_less__neg__neg,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(uminus_uminus_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ).

% less_neg_neg
tff(fact_1493_less__neg__neg,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(uminus_uminus_real,A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ).

% less_neg_neg
tff(fact_1494_less__neg__neg,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(uminus_uminus_rat,A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ).

% less_neg_neg
tff(fact_1495_less__neg__neg,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger)) ) ).

% less_neg_neg
tff(fact_1496_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_1497_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_1498_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_1499_diff__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),one_one_int) = zero_zero_int ).

% diff_numeral_special(9)
tff(fact_1500_div__self,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),A2) = one_one_rat ) ) ).

% div_self
tff(fact_1501_div__self,axiom,
    ! [A2: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),A2) = one_one_int ) ) ).

% div_self
tff(fact_1502_div__self,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),A2) = one_one_nat ) ) ).

% div_self
tff(fact_1503_div__self,axiom,
    ! [A2: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),A2) = one_one_real ) ) ).

% div_self
tff(fact_1504_div__self,axiom,
    ! [A2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A2),A2) = one_one_complex ) ) ).

% div_self
tff(fact_1505_verit__minus__simplify_I3_J,axiom,
    ! [B3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),B3) = aa_int_int(uminus_uminus_int,B3) ).

% verit_minus_simplify(3)
tff(fact_1506_verit__minus__simplify_I3_J,axiom,
    ! [B3: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),B3) = aa_real_real(uminus_uminus_real,B3) ).

% verit_minus_simplify(3)
tff(fact_1507_verit__minus__simplify_I3_J,axiom,
    ! [B3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,zero_zero_rat),B3) = aa_rat_rat(uminus_uminus_rat,B3) ).

% verit_minus_simplify(3)
tff(fact_1508_verit__minus__simplify_I3_J,axiom,
    ! [B3: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B3) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3) ).

% verit_minus_simplify(3)
tff(fact_1509_verit__minus__simplify_I3_J,axiom,
    ! [B3: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),B3) = aa_complex_complex(uminus1482373934393186551omplex,B3) ).

% verit_minus_simplify(3)
tff(fact_1510_diff__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),A2) = aa_int_int(uminus_uminus_int,A2) ).

% diff_0
tff(fact_1511_diff__0,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),A2) = aa_real_real(uminus_uminus_real,A2) ).

% diff_0
tff(fact_1512_diff__0,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,zero_zero_rat),A2) = aa_rat_rat(uminus_uminus_rat,A2) ).

% diff_0
tff(fact_1513_diff__0,axiom,
    ! [A2: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) ).

% diff_0
tff(fact_1514_diff__0,axiom,
    ! [A2: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A2) = aa_complex_complex(uminus1482373934393186551omplex,A2) ).

% diff_0
tff(fact_1515_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% zero_less_diff
tff(fact_1516_card_Oempty,axiom,
    finite_card_complex(bot_bot_set_complex) = zero_zero_nat ).

% card.empty
tff(fact_1517_card_Oempty,axiom,
    finite_card_list_nat(bot_bot_set_list_nat) = zero_zero_nat ).

% card.empty
tff(fact_1518_card_Oempty,axiom,
    finite_card_set_nat(bot_bot_set_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_1519_card_Oempty,axiom,
    finite_card_real(bot_bot_set_real) = zero_zero_nat ).

% card.empty
tff(fact_1520_card_Oempty,axiom,
    finite_card_bool(bot_bot_set_bool) = zero_zero_nat ).

% card.empty
tff(fact_1521_card_Oempty,axiom,
    aa_set_nat_nat(finite_card_nat,bot_bot_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_1522_card_Oempty,axiom,
    finite_card_int(bot_bot_set_int) = zero_zero_nat ).

% card.empty
tff(fact_1523_card_Oinfinite,axiom,
    ! [A3: set_list_nat] :
      ( ~ finite8100373058378681591st_nat(A3)
     => ( finite_card_list_nat(A3) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_1524_card_Oinfinite,axiom,
    ! [A3: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( finite_card_set_nat(A3) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_1525_card_Oinfinite,axiom,
    ! [A3: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( aa_set_nat_nat(finite_card_nat,A3) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_1526_card_Oinfinite,axiom,
    ! [A3: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A3))
     => ( finite_card_int(A3) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_1527_card_Oinfinite,axiom,
    ! [A3: set_complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( finite_card_complex(A3) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_1528_card_Oinfinite,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_1529_card_Oinfinite,axiom,
    ! [A3: set_Extended_enat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( finite121521170596916366d_enat(A3) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_1530_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% diff_is_0_eq
tff(fact_1531_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_1532_diff__Suc__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),one_one_nat) = N ).

% diff_Suc_1
tff(fact_1533_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,M) )
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_1534_negative__zle,axiom,
    ! [N: nat,M: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% negative_zle
tff(fact_1535_diff__numeral__special_I12_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% diff_numeral_special(12)
tff(fact_1536_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_1537_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_1538_diff__numeral__special_I12_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_1539_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_1540_Suc__pred,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% Suc_pred
tff(fact_1541_card__0__eq,axiom,
    ! [A3: set_list_nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( ( finite_card_list_nat(A3) = zero_zero_nat )
      <=> ( A3 = bot_bot_set_list_nat ) ) ) ).

% card_0_eq
tff(fact_1542_card__0__eq,axiom,
    ! [A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( ( finite_card_set_nat(A3) = zero_zero_nat )
      <=> ( A3 = bot_bot_set_set_nat ) ) ) ).

% card_0_eq
tff(fact_1543_card__0__eq,axiom,
    ! [A3: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( finite_card_complex(A3) = zero_zero_nat )
      <=> ( A3 = bot_bot_set_complex ) ) ) ).

% card_0_eq
tff(fact_1544_card__0__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3) = zero_zero_nat )
      <=> ( A3 = bot_bo2099793752762293965at_nat ) ) ) ).

% card_0_eq
tff(fact_1545_card__0__eq,axiom,
    ! [A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( finite121521170596916366d_enat(A3) = zero_zero_nat )
      <=> ( A3 = bot_bo7653980558646680370d_enat ) ) ) ).

% card_0_eq
tff(fact_1546_card__0__eq,axiom,
    ! [A3: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ( finite_card_real(A3) = zero_zero_nat )
      <=> ( A3 = bot_bot_set_real ) ) ) ).

% card_0_eq
tff(fact_1547_card__0__eq,axiom,
    ! [A3: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( ( finite_card_bool(A3) = zero_zero_nat )
      <=> ( A3 = bot_bot_set_bool ) ) ) ).

% card_0_eq
tff(fact_1548_card__0__eq,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( aa_set_nat_nat(finite_card_nat,A3) = zero_zero_nat )
      <=> ( A3 = bot_bot_set_nat ) ) ) ).

% card_0_eq
tff(fact_1549_card__0__eq,axiom,
    ! [A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( finite_card_int(A3) = zero_zero_nat )
      <=> ( A3 = bot_bot_set_int ) ) ) ).

% card_0_eq
tff(fact_1550_negative__zless,axiom,
    ! [N: nat,M: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% negative_zless
tff(fact_1551_finite__enumerate__mono__iff,axiom,
    ! [S: set_Extended_enat,M: nat,N: nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),finite121521170596916366d_enat(S)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),finite121521170596916366d_enat(S)))
         => ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,infini7641415182203889163d_enat(S,M)),infini7641415182203889163d_enat(S,N)))
          <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ) ) ).

% finite_enumerate_mono_iff
tff(fact_1552_finite__enumerate__mono__iff,axiom,
    ! [S: set_nat,M: nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_set_nat_nat(finite_card_nat,S)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_set_nat_nat(finite_card_nat,S)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S),M)),aa_nat_nat(infini8530281810654367211te_nat(S),N)))
          <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ) ) ).

% finite_enumerate_mono_iff
tff(fact_1553_diff__eq__diff__eq,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( ( A2 = B3 )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1554_diff__eq__diff__eq,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( ( A2 = B3 )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1555_diff__eq__diff__eq,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( ( A2 = B3 )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1556_equation__minus__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 = aa_int_int(uminus_uminus_int,B3) )
    <=> ( B3 = aa_int_int(uminus_uminus_int,A2) ) ) ).

% equation_minus_iff
tff(fact_1557_equation__minus__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( A2 = aa_real_real(uminus_uminus_real,B3) )
    <=> ( B3 = aa_real_real(uminus_uminus_real,A2) ) ) ).

% equation_minus_iff
tff(fact_1558_equation__minus__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 = aa_rat_rat(uminus_uminus_rat,B3) )
    <=> ( B3 = aa_rat_rat(uminus_uminus_rat,A2) ) ) ).

% equation_minus_iff
tff(fact_1559_equation__minus__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3) )
    <=> ( B3 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) ) ) ).

% equation_minus_iff
tff(fact_1560_equation__minus__iff,axiom,
    ! [A2: complex,B3: complex] :
      ( ( A2 = aa_complex_complex(uminus1482373934393186551omplex,B3) )
    <=> ( B3 = aa_complex_complex(uminus1482373934393186551omplex,A2) ) ) ).

% equation_minus_iff
tff(fact_1561_minus__equation__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = B3 )
    <=> ( aa_int_int(uminus_uminus_int,B3) = A2 ) ) ).

% minus_equation_iff
tff(fact_1562_minus__equation__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( aa_real_real(uminus_uminus_real,A2) = B3 )
    <=> ( aa_real_real(uminus_uminus_real,B3) = A2 ) ) ).

% minus_equation_iff
tff(fact_1563_minus__equation__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A2) = B3 )
    <=> ( aa_rat_rat(uminus_uminus_rat,B3) = A2 ) ) ).

% minus_equation_iff
tff(fact_1564_minus__equation__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = B3 )
    <=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3) = A2 ) ) ).

% minus_equation_iff
tff(fact_1565_minus__equation__iff,axiom,
    ! [A2: complex,B3: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A2) = B3 )
    <=> ( aa_complex_complex(uminus1482373934393186551omplex,B3) = A2 ) ) ).

% minus_equation_iff
tff(fact_1566_minus__diff__commute,axiom,
    ! [B3: int,A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,B3)),A2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A2)),B3) ).

% minus_diff_commute
tff(fact_1567_minus__diff__commute,axiom,
    ! [B3: real,A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,B3)),A2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A2)),B3) ).

% minus_diff_commute
tff(fact_1568_minus__diff__commute,axiom,
    ! [B3: rat,A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,B3)),A2) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,A2)),B3) ).

% minus_diff_commute
tff(fact_1569_minus__diff__commute,axiom,
    ! [B3: code_integer,A2: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3),A2) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2),B3) ).

% minus_diff_commute
tff(fact_1570_minus__diff__commute,axiom,
    ! [B3: complex,A2: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,B3)),A2) = aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),B3) ).

% minus_diff_commute
tff(fact_1571_diff__right__commute,axiom,
    ! [A2: real,C: real,B3: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C)),B3) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)),C) ).

% diff_right_commute
tff(fact_1572_diff__right__commute,axiom,
    ! [A2: rat,C: rat,B3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),C)),B3) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)),C) ).

% diff_right_commute
tff(fact_1573_diff__right__commute,axiom,
    ! [A2: nat,C: nat,B3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),C)),B3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B3)),C) ).

% diff_right_commute
tff(fact_1574_diff__right__commute,axiom,
    ! [A2: int,C: int,B3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C)),B3) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)),C) ).

% diff_right_commute
tff(fact_1575_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_1576_size__neq__size__imp__neq,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,X) != aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1577_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(X) != size_s6755466524823107622T_VEBT(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1578_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( aa_num_nat(size_size_num,X) != aa_num_nat(size_size_num,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1579_size__neq__size__imp__neq,axiom,
    ! [X: list_bool,Y: list_bool] :
      ( ( size_size_list_bool(X) != size_size_list_bool(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1580_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( size_size_list_nat(X) != size_size_list_nat(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1581_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ) ) ).

% of_nat_diff
tff(fact_1582_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ) ) ).

% of_nat_diff
tff(fact_1583_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ).

% of_nat_diff
tff(fact_1584_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% of_nat_diff
tff(fact_1585_diff__eq__diff__less__eq,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1586_diff__eq__diff__less__eq,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1587_diff__eq__diff__less__eq,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1588_diff__right__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),C))) ) ).

% diff_right_mono
tff(fact_1589_diff__right__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B3),C))) ) ).

% diff_right_mono
tff(fact_1590_diff__right__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B3),C))) ) ).

% diff_right_mono
tff(fact_1591_diff__left__mono,axiom,
    ! [B3: real,A2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B3))) ) ).

% diff_left_mono
tff(fact_1592_diff__left__mono,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B3))) ) ).

% diff_left_mono
tff(fact_1593_diff__left__mono,axiom,
    ! [B3: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B3))) ) ).

% diff_left_mono
tff(fact_1594_diff__mono,axiom,
    ! [A2: real,B3: real,D: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),D))) ) ) ).

% diff_mono
tff(fact_1595_diff__mono,axiom,
    ! [A2: rat,B3: rat,D: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B3),D))) ) ) ).

% diff_mono
tff(fact_1596_diff__mono,axiom,
    ! [A2: int,B3: int,D: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B3),D))) ) ) ).

% diff_mono
tff(fact_1597_eq__iff__diff__eq__0,axiom,
    ! [A2: real,B3: real] :
      ( ( A2 = B3 )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_1598_eq__iff__diff__eq__0,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 = B3 )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_1599_eq__iff__diff__eq__0,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 = B3 )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_1600_diff__strict__mono,axiom,
    ! [A2: real,B3: real,D: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),D))) ) ) ).

% diff_strict_mono
tff(fact_1601_diff__strict__mono,axiom,
    ! [A2: rat,B3: rat,D: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,D),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B3),D))) ) ) ).

% diff_strict_mono
tff(fact_1602_diff__strict__mono,axiom,
    ! [A2: int,B3: int,D: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,D),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B3),D))) ) ) ).

% diff_strict_mono
tff(fact_1603_diff__eq__diff__less,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_1604_diff__eq__diff__less,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_1605_diff__eq__diff__less,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_1606_diff__strict__left__mono,axiom,
    ! [B3: real,A2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B3))) ) ).

% diff_strict_left_mono
tff(fact_1607_diff__strict__left__mono,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B3))) ) ).

% diff_strict_left_mono
tff(fact_1608_diff__strict__left__mono,axiom,
    ! [B3: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B3))) ) ).

% diff_strict_left_mono
tff(fact_1609_diff__strict__right__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),C))) ) ).

% diff_strict_right_mono
tff(fact_1610_diff__strict__right__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B3),C))) ) ).

% diff_strict_right_mono
tff(fact_1611_diff__strict__right__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B3),C))) ) ).

% diff_strict_right_mono
tff(fact_1612_le__imp__neg__le,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B3)),aa_real_real(uminus_uminus_real,A2))) ) ).

% le_imp_neg_le
tff(fact_1613_le__imp__neg__le,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B3))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2))) ) ).

% le_imp_neg_le
tff(fact_1614_le__imp__neg__le,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B3)),aa_rat_rat(uminus_uminus_rat,A2))) ) ).

% le_imp_neg_le
tff(fact_1615_le__imp__neg__le,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B3)),aa_int_int(uminus_uminus_int,A2))) ) ).

% le_imp_neg_le
tff(fact_1616_minus__le__iff,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),B3))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B3)),A2)) ) ).

% minus_le_iff
tff(fact_1617_minus__le__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B3))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3)),A2)) ) ).

% minus_le_iff
tff(fact_1618_minus__le__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),B3))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B3)),A2)) ) ).

% minus_le_iff
tff(fact_1619_minus__le__iff,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),B3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B3)),A2)) ) ).

% minus_le_iff
tff(fact_1620_le__minus__iff,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(uminus_uminus_real,B3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),aa_real_real(uminus_uminus_real,A2))) ) ).

% le_minus_iff
tff(fact_1621_le__minus__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B3),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2))) ) ).

% le_minus_iff
tff(fact_1622_le__minus__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(uminus_uminus_rat,B3)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),aa_rat_rat(uminus_uminus_rat,A2))) ) ).

% le_minus_iff
tff(fact_1623_le__minus__iff,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(uminus_uminus_int,B3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),aa_int_int(uminus_uminus_int,A2))) ) ).

% le_minus_iff
tff(fact_1624_verit__negate__coefficient_I2_J,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B3)),aa_int_int(uminus_uminus_int,A2))) ) ).

% verit_negate_coefficient(2)
tff(fact_1625_verit__negate__coefficient_I2_J,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B3)),aa_real_real(uminus_uminus_real,A2))) ) ).

% verit_negate_coefficient(2)
tff(fact_1626_verit__negate__coefficient_I2_J,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B3)),aa_rat_rat(uminus_uminus_rat,A2))) ) ).

% verit_negate_coefficient(2)
tff(fact_1627_verit__negate__coefficient_I2_J,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B3))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2))) ) ).

% verit_negate_coefficient(2)
tff(fact_1628_less__minus__iff,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(uminus_uminus_int,B3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),aa_int_int(uminus_uminus_int,A2))) ) ).

% less_minus_iff
tff(fact_1629_less__minus__iff,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(uminus_uminus_real,B3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),aa_real_real(uminus_uminus_real,A2))) ) ).

% less_minus_iff
tff(fact_1630_less__minus__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(uminus_uminus_rat,B3)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(uminus_uminus_rat,A2))) ) ).

% less_minus_iff
tff(fact_1631_less__minus__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B3),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2))) ) ).

% less_minus_iff
tff(fact_1632_minus__less__iff,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A2)),B3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B3)),A2)) ) ).

% minus_less_iff
tff(fact_1633_minus__less__iff,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A2)),B3))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B3)),A2)) ) ).

% minus_less_iff
tff(fact_1634_minus__less__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A2)),B3))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B3)),A2)) ) ).

% minus_less_iff
tff(fact_1635_minus__less__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B3))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3)),A2)) ) ).

% minus_less_iff
tff(fact_1636_zero__induct__lemma,axiom,
    ! [P: fun_nat_bool,K: nat,I: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(P,N2)) )
       => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I))) ) ) ).

% zero_induct_lemma
tff(fact_1637_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M) = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
tff(fact_1638_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_1639_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ) ).

% diff_less_mono2
tff(fact_1640_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),N)),K)) ) ).

% less_imp_diff_less
tff(fact_1641_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ).

% diff_le_mono2
tff(fact_1642_le__diff__iff_H,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),C))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B3)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2)) ) ) ) ).

% le_diff_iff'
tff(fact_1643_diff__le__self,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ).

% diff_le_self
tff(fact_1644_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),L))) ) ).

% diff_le_mono
tff(fact_1645_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( 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,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_1646_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% le_diff_iff
tff(fact_1647_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K) )
        <=> ( M = N ) ) ) ) ).

% eq_diff_iff
tff(fact_1648_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] : vEBT_vebt_delete(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary),Uu) = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary) ).

% vebt_delete.simps(4)
tff(fact_1649_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),X)) ).

% vebt_member.simps(2)
tff(fact_1650_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy))) ).

% VEBT_internal.minNull.simps(4)
tff(fact_1651_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb,Vc))) ).

% VEBT_internal.minNull.simps(5)
tff(fact_1652_verit__less__mono__div__int2,axiom,
    ! [A3: int,B2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A3),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,N)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_1653_infinite__arbitrarily__large,axiom,
    ! [A3: set_list_nat,N: nat] :
      ( ~ finite8100373058378681591st_nat(A3)
     => ? [B7: set_list_nat] :
          ( finite8100373058378681591st_nat(B7)
          & ( finite_card_list_nat(B7) = N )
          & ord_le6045566169113846134st_nat(B7,A3) ) ) ).

% infinite_arbitrarily_large
tff(fact_1654_infinite__arbitrarily__large,axiom,
    ! [A3: set_set_nat,N: nat] :
      ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ? [B7: set_set_nat] :
          ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B7))
          & ( finite_card_set_nat(B7) = N )
          & pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(B7),A3)) ) ) ).

% infinite_arbitrarily_large
tff(fact_1655_infinite__arbitrarily__large,axiom,
    ! [A3: set_nat,N: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ? [B7: set_nat] :
          ( pp(aa_set_nat_bool(finite_finite_nat,B7))
          & ( aa_set_nat_nat(finite_card_nat,B7) = N )
          & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),A3)) ) ) ).

% infinite_arbitrarily_large
tff(fact_1656_infinite__arbitrarily__large,axiom,
    ! [A3: set_complex,N: nat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ? [B7: set_complex] :
          ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B7))
          & ( finite_card_complex(B7) = N )
          & pp(aa_set_complex_bool(ord_le211207098394363844omplex(B7),A3)) ) ) ).

% infinite_arbitrarily_large
tff(fact_1657_infinite__arbitrarily__large,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ? [B7: set_Pr1261947904930325089at_nat] :
          ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B7))
          & ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B7) = N )
          & pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(B7),A3)) ) ) ).

% infinite_arbitrarily_large
tff(fact_1658_infinite__arbitrarily__large,axiom,
    ! [A3: set_Extended_enat,N: nat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ? [B7: set_Extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B7))
          & ( finite121521170596916366d_enat(B7) = N )
          & pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B7),A3)) ) ) ).

% infinite_arbitrarily_large
tff(fact_1659_infinite__arbitrarily__large,axiom,
    ! [A3: set_int,N: nat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A3))
     => ? [B7: set_int] :
          ( pp(aa_set_int_bool(finite_finite_int,B7))
          & ( finite_card_int(B7) = N )
          & pp(aa_set_int_bool(ord_less_eq_set_int(B7),A3)) ) ) ).

% infinite_arbitrarily_large
tff(fact_1660_card__subset__eq,axiom,
    ! [B2: set_list_nat,A3: set_list_nat] :
      ( finite8100373058378681591st_nat(B2)
     => ( ord_le6045566169113846134st_nat(A3,B2)
       => ( ( finite_card_list_nat(A3) = finite_card_list_nat(B2) )
         => ( A3 = B2 ) ) ) ) ).

% card_subset_eq
tff(fact_1661_card__subset__eq,axiom,
    ! [B2: set_set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B2))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),B2))
       => ( ( finite_card_set_nat(A3) = finite_card_set_nat(B2) )
         => ( A3 = B2 ) ) ) ) ).

% card_subset_eq
tff(fact_1662_card__subset__eq,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
       => ( ( aa_set_nat_nat(finite_card_nat,A3) = aa_set_nat_nat(finite_card_nat,B2) )
         => ( A3 = B2 ) ) ) ) ).

% card_subset_eq
tff(fact_1663_card__subset__eq,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),B2))
       => ( ( finite_card_complex(A3) = finite_card_complex(B2) )
         => ( A3 = B2 ) ) ) ) ).

% card_subset_eq
tff(fact_1664_card__subset__eq,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),B2))
       => ( ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3) = aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B2) )
         => ( A3 = B2 ) ) ) ) ).

% card_subset_eq
tff(fact_1665_card__subset__eq,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),B2))
       => ( ( finite121521170596916366d_enat(A3) = finite121521170596916366d_enat(B2) )
         => ( A3 = B2 ) ) ) ) ).

% card_subset_eq
tff(fact_1666_card__subset__eq,axiom,
    ! [B2: set_int,A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
       => ( ( finite_card_int(A3) = finite_card_int(B2) )
         => ( A3 = B2 ) ) ) ) ).

% card_subset_eq
tff(fact_1667_le__iff__diff__le__0,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)),zero_zero_real)) ) ).

% le_iff_diff_le_0
tff(fact_1668_le__iff__diff__le__0,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)),zero_zero_rat)) ) ).

% le_iff_diff_le_0
tff(fact_1669_le__iff__diff__le__0,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)),zero_zero_int)) ) ).

% le_iff_diff_le_0
tff(fact_1670_less__iff__diff__less__0,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)),zero_zero_real)) ) ).

% less_iff_diff_less_0
tff(fact_1671_less__iff__diff__less__0,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)),zero_zero_rat)) ) ).

% less_iff_diff_less_0
tff(fact_1672_less__iff__diff__less__0,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)),zero_zero_int)) ) ).

% less_iff_diff_less_0
tff(fact_1673_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A3: set_real,R2: fun_re1255401033344146673l_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( ! [A: real] :
            ( pp(aa_set_real_bool(member_real(A),A3))
           => ? [B8: real] :
                ( pp(aa_set_real_bool(member_real(B8),B2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,A),B8)) ) )
       => ( ! [A1: real,A22: real,B: real] :
              ( pp(aa_set_real_bool(member_real(A1),A3))
             => ( pp(aa_set_real_bool(member_real(A22),A3))
               => ( pp(aa_set_real_bool(member_real(B),B2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,A1),B))
                   => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,A22),B))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(A3)),finite_card_real(B2))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_1674_card__le__if__inj__on__rel,axiom,
    ! [B2: set_bool,A3: set_real,R2: fun_re3316061561565328172l_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,B2))
     => ( ! [A: real] :
            ( pp(aa_set_real_bool(member_real(A),A3))
           => ? [B8: bool] :
                ( pp(aa_set_bool_bool(member_bool(B8),B2))
                & pp(aa_bool_bool(aa_rea6668032140610767853l_bool(R2,A),B8)) ) )
       => ( ! [A1: real,A22: real,B: bool] :
              ( pp(aa_set_real_bool(member_real(A1),A3))
             => ( pp(aa_set_real_bool(member_real(A22),A3))
               => ( pp(aa_set_bool_bool(member_bool(B),B2))
                 => ( pp(aa_bool_bool(aa_rea6668032140610767853l_bool(R2,A1),B))
                   => ( pp(aa_bool_bool(aa_rea6668032140610767853l_bool(R2,A22),B))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(A3)),finite_card_bool(B2))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_1675_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A3: set_bool,R2: fun_bo2025495424286513072l_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( ! [A: bool] :
            ( pp(aa_set_bool_bool(member_bool(A),A3))
           => ? [B8: real] :
                ( pp(aa_set_real_bool(member_real(B8),B2))
                & pp(aa_real_bool(aa_boo2589871674041770537l_bool(R2,A),B8)) ) )
       => ( ! [A1: bool,A22: bool,B: real] :
              ( pp(aa_set_bool_bool(member_bool(A1),A3))
             => ( pp(aa_set_bool_bool(member_bool(A22),A3))
               => ( pp(aa_set_real_bool(member_real(B),B2))
                 => ( pp(aa_real_bool(aa_boo2589871674041770537l_bool(R2,A1),B))
                   => ( pp(aa_real_bool(aa_boo2589871674041770537l_bool(R2,A22),B))
                     => ( pp(A1)
                      <=> pp(A22) ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_bool(A3)),finite_card_real(B2))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_1676_card__le__if__inj__on__rel,axiom,
    ! [B2: set_bool,A3: set_bool,R2: fun_bo1841576699660343469l_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,B2))
     => ( ! [A: bool] :
            ( pp(aa_set_bool_bool(member_bool(A),A3))
           => ? [B8: bool] :
                ( pp(aa_set_bool_bool(member_bool(B8),B2))
                & pp(aa_bool_bool(aa_boo8970175072276269406l_bool(R2,A),B8)) ) )
       => ( ! [A1: bool,A22: bool,B: bool] :
              ( pp(aa_set_bool_bool(member_bool(A1),A3))
             => ( pp(aa_set_bool_bool(member_bool(A22),A3))
               => ( pp(aa_set_bool_bool(member_bool(B),B2))
                 => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(R2,A1),B))
                   => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(R2,A22),B))
                     => ( pp(A1)
                      <=> pp(A22) ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_bool(A3)),finite_card_bool(B2))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_1677_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A3: set_complex,R2: fun_co2286642472385812463l_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( ! [A: complex] :
            ( pp(aa_set_complex_bool(member_complex(A),A3))
           => ? [B8: real] :
                ( pp(aa_set_real_bool(member_real(B8),B2))
                & pp(aa_real_bool(aa_com8943241093633526360l_bool(R2,A),B8)) ) )
       => ( ! [A1: complex,A22: complex,B: real] :
              ( pp(aa_set_complex_bool(member_complex(A1),A3))
             => ( pp(aa_set_complex_bool(member_complex(A22),A3))
               => ( pp(aa_set_real_bool(member_real(B),B2))
                 => ( pp(aa_real_bool(aa_com8943241093633526360l_bool(R2,A1),B))
                   => ( pp(aa_real_bool(aa_com8943241093633526360l_bool(R2,A22),B))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(A3)),finite_card_real(B2))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_1678_card__le__if__inj__on__rel,axiom,
    ! [B2: set_bool,A3: set_complex,R2: fun_co7455972192742496430l_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,B2))
     => ( ! [A: complex] :
            ( pp(aa_set_complex_bool(member_complex(A),A3))
           => ? [B8: bool] :
                ( pp(aa_set_bool_bool(member_bool(B8),B2))
                & pp(aa_bool_bool(aa_com4332222969190720879l_bool(R2,A),B8)) ) )
       => ( ! [A1: complex,A22: complex,B: bool] :
              ( pp(aa_set_complex_bool(member_complex(A1),A3))
             => ( pp(aa_set_complex_bool(member_complex(A22),A3))
               => ( pp(aa_set_bool_bool(member_bool(B),B2))
                 => ( pp(aa_bool_bool(aa_com4332222969190720879l_bool(R2,A1),B))
                   => ( pp(aa_bool_bool(aa_com4332222969190720879l_bool(R2,A22),B))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(A3)),finite_card_bool(B2))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_1679_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A3: set_nat,R2: fun_na6801867027684862029l_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( ! [A: nat] :
            ( pp(aa_set_nat_bool(member_nat(A),A3))
           => ? [B8: real] :
                ( pp(aa_set_real_bool(member_real(B8),B2))
                & pp(aa_real_bool(aa_nat_fun_real_bool(R2,A),B8)) ) )
       => ( ! [A1: nat,A22: nat,B: real] :
              ( pp(aa_set_nat_bool(member_nat(A1),A3))
             => ( pp(aa_set_nat_bool(member_nat(A22),A3))
               => ( pp(aa_set_real_bool(member_real(B),B2))
                 => ( pp(aa_real_bool(aa_nat_fun_real_bool(R2,A1),B))
                   => ( pp(aa_real_bool(aa_nat_fun_real_bool(R2,A22),B))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A3)),finite_card_real(B2))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_1680_card__le__if__inj__on__rel,axiom,
    ! [B2: set_bool,A3: set_nat,R2: fun_na9137082442855928016l_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,B2))
     => ( ! [A: nat] :
            ( pp(aa_set_nat_bool(member_nat(A),A3))
           => ? [B8: bool] :
                ( pp(aa_set_bool_bool(member_bool(B8),B2))
                & pp(aa_bool_bool(aa_nat_fun_bool_bool(R2,A),B8)) ) )
       => ( ! [A1: nat,A22: nat,B: bool] :
              ( pp(aa_set_nat_bool(member_nat(A1),A3))
             => ( pp(aa_set_nat_bool(member_nat(A22),A3))
               => ( pp(aa_set_bool_bool(member_bool(B),B2))
                 => ( pp(aa_bool_bool(aa_nat_fun_bool_bool(R2,A1),B))
                   => ( pp(aa_bool_bool(aa_nat_fun_bool_bool(R2,A22),B))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A3)),finite_card_bool(B2))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_1681_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A3: set_int,R2: fun_in3970322703178350321l_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( ! [A: int] :
            ( pp(aa_set_int_bool(member_int(A),A3))
           => ? [B8: real] :
                ( pp(aa_set_real_bool(member_real(B8),B2))
                & pp(aa_real_bool(aa_int_fun_real_bool(R2,A),B8)) ) )
       => ( ! [A1: int,A22: int,B: real] :
              ( pp(aa_set_int_bool(member_int(A1),A3))
             => ( pp(aa_set_int_bool(member_int(A22),A3))
               => ( pp(aa_set_real_bool(member_real(B),B2))
                 => ( pp(aa_real_bool(aa_int_fun_real_bool(R2,A1),B))
                   => ( pp(aa_real_bool(aa_int_fun_real_bool(R2,A22),B))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(A3)),finite_card_real(B2))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_1682_card__le__if__inj__on__rel,axiom,
    ! [B2: set_bool,A3: set_int,R2: fun_in3865155034416861484l_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,B2))
     => ( ! [A: int] :
            ( pp(aa_set_int_bool(member_int(A),A3))
           => ? [B8: bool] :
                ( pp(aa_set_bool_bool(member_bool(B8),B2))
                & pp(aa_bool_bool(aa_int_fun_bool_bool(R2,A),B8)) ) )
       => ( ! [A1: int,A22: int,B: bool] :
              ( pp(aa_set_int_bool(member_int(A1),A3))
             => ( pp(aa_set_int_bool(member_int(A22),A3))
               => ( pp(aa_set_bool_bool(member_bool(B),B2))
                 => ( pp(aa_bool_bool(aa_int_fun_bool_bool(R2,A1),B))
                   => ( pp(aa_bool_bool(aa_int_fun_bool_bool(R2,A22),B))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(A3)),finite_card_bool(B2))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_1683_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% le_minus_one_simps(4)
tff(fact_1684_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% le_minus_one_simps(4)
tff(fact_1685_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% le_minus_one_simps(4)
tff(fact_1686_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% le_minus_one_simps(4)
tff(fact_1687_le__minus__one__simps_I2_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).

% le_minus_one_simps(2)
tff(fact_1688_le__minus__one__simps_I2_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer)) ).

% le_minus_one_simps(2)
tff(fact_1689_le__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat)) ).

% le_minus_one_simps(2)
tff(fact_1690_le__minus__one__simps_I2_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).

% le_minus_one_simps(2)
tff(fact_1691_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_1692_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_1693_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_1694_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_1695_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_1696_less__minus__one__simps_I2_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).

% less_minus_one_simps(2)
tff(fact_1697_less__minus__one__simps_I2_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).

% less_minus_one_simps(2)
tff(fact_1698_less__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat)) ).

% less_minus_one_simps(2)
tff(fact_1699_less__minus__one__simps_I2_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer)) ).

% less_minus_one_simps(2)
tff(fact_1700_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% less_minus_one_simps(4)
tff(fact_1701_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% less_minus_one_simps(4)
tff(fact_1702_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% less_minus_one_simps(4)
tff(fact_1703_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% less_minus_one_simps(4)
tff(fact_1704_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(suc,M))) ).

% diff_less_Suc
tff(fact_1705_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ).

% Suc_diff_Suc
tff(fact_1706_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ) ) ).

% diff_less
tff(fact_1707_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ) ).

% Suc_diff_le
tff(fact_1708_diff__less__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B3),C))) ) ) ).

% diff_less_mono
tff(fact_1709_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ) ).

% less_diff_iff
tff(fact_1710_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N) ).

% diff_Suc_eq_diff_pred
tff(fact_1711_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% not_int_zless_negative
tff(fact_1712_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),X)) ).

% vebt_member.simps(3)
tff(fact_1713_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)),Uz)) ).

% VEBT_internal.membermima.simps(2)
tff(fact_1714_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = none_nat ).

% vebt_mint.simps(2)
tff(fact_1715_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = none_nat ).

% vebt_maxt.simps(2)
tff(fact_1716_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_1717_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( ( X != vEBT_Leaf(fFalse,fFalse) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_1718_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
      <=> pp(Y) )
     => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
         => ~ pp(Y) )
       => ( ( ? [Uv2: bool] : X = vEBT_Leaf(fTrue,Uv2)
           => pp(Y) )
         => ( ( ? [Uu2: bool] : X = vEBT_Leaf(Uu2,fTrue)
             => pp(Y) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
               => ~ pp(Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)
                 => pp(Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_1719_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero_nat,Uv,Uw)),Ux)) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_1720_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] : vEBT_vebt_pred(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz,Va),Vb) = none_nat ).

% vebt_pred.simps(4)
tff(fact_1721_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] : vEBT_vebt_succ(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy,Uz),Va) = none_nat ).

% vebt_succ.simps(3)
tff(fact_1722_frac__ge__0,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,X))) ).

% frac_ge_0
tff(fact_1723_frac__ge__0,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X))) ).

% frac_ge_0
tff(fact_1724_frac__lt__1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(archim2898591450579166408c_real,X)),one_one_real)) ).

% frac_lt_1
tff(fact_1725_frac__lt__1,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(archimedean_frac_rat,X)),one_one_rat)) ).

% frac_lt_1
tff(fact_1726_card__eq__0__iff,axiom,
    ! [A3: set_list_nat] :
      ( ( finite_card_list_nat(A3) = zero_zero_nat )
    <=> ( ( A3 = bot_bot_set_list_nat )
        | ~ finite8100373058378681591st_nat(A3) ) ) ).

% card_eq_0_iff
tff(fact_1727_card__eq__0__iff,axiom,
    ! [A3: set_set_nat] :
      ( ( finite_card_set_nat(A3) = zero_zero_nat )
    <=> ( ( A3 = bot_bot_set_set_nat )
        | ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3)) ) ) ).

% card_eq_0_iff
tff(fact_1728_card__eq__0__iff,axiom,
    ! [A3: set_complex] :
      ( ( finite_card_complex(A3) = zero_zero_nat )
    <=> ( ( A3 = bot_bot_set_complex )
        | ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A3)) ) ) ).

% card_eq_0_iff
tff(fact_1729_card__eq__0__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3) = zero_zero_nat )
    <=> ( ( A3 = bot_bo2099793752762293965at_nat )
        | ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3)) ) ) ).

% card_eq_0_iff
tff(fact_1730_card__eq__0__iff,axiom,
    ! [A3: set_Extended_enat] :
      ( ( finite121521170596916366d_enat(A3) = zero_zero_nat )
    <=> ( ( A3 = bot_bo7653980558646680370d_enat )
        | ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3)) ) ) ).

% card_eq_0_iff
tff(fact_1731_card__eq__0__iff,axiom,
    ! [A3: set_real] :
      ( ( finite_card_real(A3) = zero_zero_nat )
    <=> ( ( A3 = bot_bot_set_real )
        | ~ pp(aa_set_real_bool(finite_finite_real,A3)) ) ) ).

% card_eq_0_iff
tff(fact_1732_card__eq__0__iff,axiom,
    ! [A3: set_bool] :
      ( ( finite_card_bool(A3) = zero_zero_nat )
    <=> ( ( A3 = bot_bot_set_bool )
        | ~ pp(aa_set_bool_bool(finite_finite_bool,A3)) ) ) ).

% card_eq_0_iff
tff(fact_1733_card__eq__0__iff,axiom,
    ! [A3: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A3) = zero_zero_nat )
    <=> ( ( A3 = bot_bot_set_nat )
        | ~ pp(aa_set_nat_bool(finite_finite_nat,A3)) ) ) ).

% card_eq_0_iff
tff(fact_1734_card__eq__0__iff,axiom,
    ! [A3: set_int] :
      ( ( finite_card_int(A3) = zero_zero_nat )
    <=> ( ( A3 = bot_bot_set_int )
        | ~ pp(aa_set_int_bool(finite_finite_int,A3)) ) ) ).

% card_eq_0_iff
tff(fact_1735_card__ge__0__finite,axiom,
    ! [A3: set_list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_list_nat(A3)))
     => finite8100373058378681591st_nat(A3) ) ).

% card_ge_0_finite
tff(fact_1736_card__ge__0__finite,axiom,
    ! [A3: set_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_set_nat(A3)))
     => pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3)) ) ).

% card_ge_0_finite
tff(fact_1737_card__ge__0__finite,axiom,
    ! [A3: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A3)))
     => pp(aa_set_nat_bool(finite_finite_nat,A3)) ) ).

% card_ge_0_finite
tff(fact_1738_card__ge__0__finite,axiom,
    ! [A3: set_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_int(A3)))
     => pp(aa_set_int_bool(finite_finite_int,A3)) ) ).

% card_ge_0_finite
tff(fact_1739_card__ge__0__finite,axiom,
    ! [A3: set_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_complex(A3)))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,A3)) ) ).

% card_ge_0_finite
tff(fact_1740_card__ge__0__finite,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)))
     => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3)) ) ).

% card_ge_0_finite
tff(fact_1741_card__ge__0__finite,axiom,
    ! [A3: set_Extended_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite121521170596916366d_enat(A3)))
     => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3)) ) ).

% card_ge_0_finite
tff(fact_1742_card__mono,axiom,
    ! [B2: set_list_nat,A3: set_list_nat] :
      ( finite8100373058378681591st_nat(B2)
     => ( ord_le6045566169113846134st_nat(A3,B2)
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(A3)),finite_card_list_nat(B2))) ) ) ).

% card_mono
tff(fact_1743_card__mono,axiom,
    ! [B2: set_set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B2))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(A3)),finite_card_set_nat(B2))) ) ) ).

% card_mono
tff(fact_1744_card__mono,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,B2))) ) ) ).

% card_mono
tff(fact_1745_card__mono,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(A3)),finite_card_complex(B2))) ) ) ).

% card_mono
tff(fact_1746_card__mono,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B2))) ) ) ).

% card_mono
tff(fact_1747_card__mono,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite121521170596916366d_enat(A3)),finite121521170596916366d_enat(B2))) ) ) ).

% card_mono
tff(fact_1748_card__mono,axiom,
    ! [B2: set_int,A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(A3)),finite_card_int(B2))) ) ) ).

% card_mono
tff(fact_1749_card__seteq,axiom,
    ! [B2: set_list_nat,A3: set_list_nat] :
      ( finite8100373058378681591st_nat(B2)
     => ( ord_le6045566169113846134st_nat(A3,B2)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(B2)),finite_card_list_nat(A3)))
         => ( A3 = B2 ) ) ) ) ).

% card_seteq
tff(fact_1750_card__seteq,axiom,
    ! [B2: set_set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B2))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(B2)),finite_card_set_nat(A3)))
         => ( A3 = B2 ) ) ) ) ).

% card_seteq
tff(fact_1751_card__seteq,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,B2)),aa_set_nat_nat(finite_card_nat,A3)))
         => ( A3 = B2 ) ) ) ) ).

% card_seteq
tff(fact_1752_card__seteq,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(B2)),finite_card_complex(A3)))
         => ( A3 = B2 ) ) ) ) ).

% card_seteq
tff(fact_1753_card__seteq,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B2)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)))
         => ( A3 = B2 ) ) ) ) ).

% card_seteq
tff(fact_1754_card__seteq,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite121521170596916366d_enat(B2)),finite121521170596916366d_enat(A3)))
         => ( A3 = B2 ) ) ) ) ).

% card_seteq
tff(fact_1755_card__seteq,axiom,
    ! [B2: set_int,A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(B2)),finite_card_int(A3)))
         => ( A3 = B2 ) ) ) ) ).

% card_seteq
tff(fact_1756_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_list_nat,C2: nat] :
      ( ! [G: set_list_nat] :
          ( ord_le6045566169113846134st_nat(G,F2)
         => ( finite8100373058378681591st_nat(G)
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(G)),C2)) ) )
     => ( finite8100373058378681591st_nat(F2)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(F2)),C2)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_1757_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_set_nat,C2: nat] :
      ( ! [G: set_set_nat] :
          ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(G),F2))
         => ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,G))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(G)),C2)) ) )
     => ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(F2)),C2)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_1758_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_nat,C2: nat] :
      ( ! [G: set_nat] :
          ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,G),F2))
         => ( pp(aa_set_nat_bool(finite_finite_nat,G))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,G)),C2)) ) )
     => ( pp(aa_set_nat_bool(finite_finite_nat,F2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,F2)),C2)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_1759_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_complex,C2: nat] :
      ( ! [G: set_complex] :
          ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(G),F2))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,G))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(G)),C2)) ) )
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(F2)),C2)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_1760_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,C2: nat] :
      ( ! [G: set_Pr1261947904930325089at_nat] :
          ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(G),F2))
         => ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,G))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,G)),C2)) ) )
     => ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,F2)),C2)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_1761_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_Extended_enat,C2: nat] :
      ( ! [G: set_Extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(G),F2))
         => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,G))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite121521170596916366d_enat(G)),C2)) ) )
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite121521170596916366d_enat(F2)),C2)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_1762_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_int,C2: nat] :
      ( ! [G: set_int] :
          ( pp(aa_set_int_bool(ord_less_eq_set_int(G),F2))
         => ( pp(aa_set_int_bool(finite_finite_int,G))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(G)),C2)) ) )
     => ( pp(aa_set_int_bool(finite_finite_int,F2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(F2)),C2)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_1763_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_list_nat(S)))
     => ~ ! [T4: set_list_nat] :
            ( ord_le6045566169113846134st_nat(T4,S)
           => ( ( finite_card_list_nat(T4) = N )
             => ~ finite8100373058378681591st_nat(T4) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_1764_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_set_nat(S)))
     => ~ ! [T4: set_set_nat] :
            ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(T4),S))
           => ( ( finite_card_set_nat(T4) = N )
             => ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,T4)) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_1765_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_set_nat_nat(finite_card_nat,S)))
     => ~ ! [T4: set_nat] :
            ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,T4),S))
           => ( ( aa_set_nat_nat(finite_card_nat,T4) = N )
             => ~ pp(aa_set_nat_bool(finite_finite_nat,T4)) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_1766_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_complex(S)))
     => ~ ! [T4: set_complex] :
            ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(T4),S))
           => ( ( finite_card_complex(T4) = N )
             => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,T4)) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_1767_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_Pr1261947904930325089at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,S)))
     => ~ ! [T4: set_Pr1261947904930325089at_nat] :
            ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(T4),S))
           => ( ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,T4) = N )
             => ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,T4)) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_1768_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_Extended_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite121521170596916366d_enat(S)))
     => ~ ! [T4: set_Extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(T4),S))
           => ( ( finite121521170596916366d_enat(T4) = N )
             => ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4)) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_1769_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_int(S)))
     => ~ ! [T4: set_int] :
            ( pp(aa_set_int_bool(ord_less_eq_set_int(T4),S))
           => ( ( finite_card_int(T4) = N )
             => ~ pp(aa_set_int_bool(finite_finite_int,T4)) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_1770_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% le_minus_one_simps(3)
tff(fact_1771_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% le_minus_one_simps(3)
tff(fact_1772_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% le_minus_one_simps(3)
tff(fact_1773_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% le_minus_one_simps(3)
tff(fact_1774_le__minus__one__simps_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).

% le_minus_one_simps(1)
tff(fact_1775_le__minus__one__simps_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger)) ).

% le_minus_one_simps(1)
tff(fact_1776_le__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat)) ).

% le_minus_one_simps(1)
tff(fact_1777_le__minus__one__simps_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).

% le_minus_one_simps(1)
tff(fact_1778_less__minus__one__simps_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).

% less_minus_one_simps(1)
tff(fact_1779_less__minus__one__simps_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).

% less_minus_one_simps(1)
tff(fact_1780_less__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat)) ).

% less_minus_one_simps(1)
tff(fact_1781_less__minus__one__simps_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger)) ).

% less_minus_one_simps(1)
tff(fact_1782_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% less_minus_one_simps(3)
tff(fact_1783_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% less_minus_one_simps(3)
tff(fact_1784_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% less_minus_one_simps(3)
tff(fact_1785_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% less_minus_one_simps(3)
tff(fact_1786_psubset__card__mono,axiom,
    ! [B2: set_list_nat,A3: set_list_nat] :
      ( finite8100373058378681591st_nat(B2)
     => ( ord_le1190675801316882794st_nat(A3,B2)
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_list_nat(A3)),finite_card_list_nat(B2))) ) ) ).

% psubset_card_mono
tff(fact_1787_psubset__card__mono,axiom,
    ! [B2: set_set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B2))
     => ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A3),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_set_nat(A3)),finite_card_set_nat(B2))) ) ) ).

% psubset_card_mono
tff(fact_1788_psubset__card__mono,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A3),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,B2))) ) ) ).

% psubset_card_mono
tff(fact_1789_psubset__card__mono,axiom,
    ! [B2: set_int,A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_set_int(A3),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_int(A3)),finite_card_int(B2))) ) ) ).

% psubset_card_mono
tff(fact_1790_psubset__card__mono,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_less_set_complex(A3),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_complex(A3)),finite_card_complex(B2))) ) ) ).

% psubset_card_mono
tff(fact_1791_psubset__card__mono,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
     => ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A3),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B2))) ) ) ).

% psubset_card_mono
tff(fact_1792_psubset__card__mono,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(A3),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite121521170596916366d_enat(A3)),finite121521170596916366d_enat(B2))) ) ) ).

% psubset_card_mono
tff(fact_1793_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,I))),N)) ) ).

% diff_Suc_less
tff(fact_1794_finite__enumerate__in__set,axiom,
    ! [S: set_Extended_enat,N: nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),finite121521170596916366d_enat(S)))
       => pp(aa_set3347977610044113549t_bool(member_Extended_enat(infini7641415182203889163d_enat(S,N)),S)) ) ) ).

% finite_enumerate_in_set
tff(fact_1795_finite__enumerate__in__set,axiom,
    ! [S: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_set_nat_nat(finite_card_nat,S)))
       => pp(aa_set_nat_bool(member_nat(aa_nat_nat(infini8530281810654367211te_nat(S),N)),S)) ) ) ).

% finite_enumerate_in_set
tff(fact_1796_finite__enumerate__Ex,axiom,
    ! [S: set_Extended_enat,S2: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(S2),S))
       => ? [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),finite121521170596916366d_enat(S)))
            & ( infini7641415182203889163d_enat(S,N2) = S2 ) ) ) ) ).

% finite_enumerate_Ex
tff(fact_1797_finite__enumerate__Ex,axiom,
    ! [S: set_nat,S2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_nat_bool(member_nat(S2),S))
       => ? [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_set_nat_nat(finite_card_nat,S)))
            & ( aa_nat_nat(infini8530281810654367211te_nat(S),N2) = S2 ) ) ) ) ).

% finite_enumerate_Ex
tff(fact_1798_finite__enum__ext,axiom,
    ! [X6: set_Extended_enat,Y5: set_Extended_enat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),finite121521170596916366d_enat(X6)))
         => ( infini7641415182203889163d_enat(X6,I2) = infini7641415182203889163d_enat(Y5,I2) ) )
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,X6))
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,Y5))
         => ( ( finite121521170596916366d_enat(X6) = finite121521170596916366d_enat(Y5) )
           => ( X6 = Y5 ) ) ) ) ) ).

% finite_enum_ext
tff(fact_1799_finite__enum__ext,axiom,
    ! [X6: set_nat,Y5: set_nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_set_nat_nat(finite_card_nat,X6)))
         => ( aa_nat_nat(infini8530281810654367211te_nat(X6),I2) = aa_nat_nat(infini8530281810654367211te_nat(Y5),I2) ) )
     => ( pp(aa_set_nat_bool(finite_finite_nat,X6))
       => ( pp(aa_set_nat_bool(finite_finite_nat,Y5))
         => ( ( aa_set_nat_nat(finite_card_nat,X6) = aa_set_nat_nat(finite_card_nat,Y5) )
           => ( X6 = Y5 ) ) ) ) ) ).

% finite_enum_ext
tff(fact_1800_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ) ) ).

% int_cases4
tff(fact_1801_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),X)) ).

% vebt_member.simps(4)
tff(fact_1802_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))))
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_1803_negative__zle__0,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),zero_zero_int)) ).

% negative_zle_0
tff(fact_1804_nonpos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ~ ! [N2: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% nonpos_int_cases
tff(fact_1805_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve),Vf) = none_nat ).

% vebt_pred.simps(5)
tff(fact_1806_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc,Vd),Ve) = none_nat ).

% vebt_succ.simps(4)
tff(fact_1807_card__gt__0__iff,axiom,
    ! [A3: set_list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_list_nat(A3)))
    <=> ( ( A3 != bot_bot_set_list_nat )
        & finite8100373058378681591st_nat(A3) ) ) ).

% card_gt_0_iff
tff(fact_1808_card__gt__0__iff,axiom,
    ! [A3: set_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_set_nat(A3)))
    <=> ( ( A3 != bot_bot_set_set_nat )
        & pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3)) ) ) ).

% card_gt_0_iff
tff(fact_1809_card__gt__0__iff,axiom,
    ! [A3: set_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_complex(A3)))
    <=> ( ( A3 != bot_bot_set_complex )
        & pp(aa_set_complex_bool(finite3207457112153483333omplex,A3)) ) ) ).

% card_gt_0_iff
tff(fact_1810_card__gt__0__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)))
    <=> ( ( A3 != bot_bo2099793752762293965at_nat )
        & pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3)) ) ) ).

% card_gt_0_iff
tff(fact_1811_card__gt__0__iff,axiom,
    ! [A3: set_Extended_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite121521170596916366d_enat(A3)))
    <=> ( ( A3 != bot_bo7653980558646680370d_enat )
        & pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3)) ) ) ).

% card_gt_0_iff
tff(fact_1812_card__gt__0__iff,axiom,
    ! [A3: set_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_real(A3)))
    <=> ( ( A3 != bot_bot_set_real )
        & pp(aa_set_real_bool(finite_finite_real,A3)) ) ) ).

% card_gt_0_iff
tff(fact_1813_card__gt__0__iff,axiom,
    ! [A3: set_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_bool(A3)))
    <=> ( ( A3 != bot_bot_set_bool )
        & pp(aa_set_bool_bool(finite_finite_bool,A3)) ) ) ).

% card_gt_0_iff
tff(fact_1814_card__gt__0__iff,axiom,
    ! [A3: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A3)))
    <=> ( ( A3 != bot_bot_set_nat )
        & pp(aa_set_nat_bool(finite_finite_nat,A3)) ) ) ).

% card_gt_0_iff
tff(fact_1815_card__gt__0__iff,axiom,
    ! [A3: set_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_int(A3)))
    <=> ( ( A3 != bot_bot_set_int )
        & pp(aa_set_int_bool(finite_finite_int,A3)) ) ) ).

% card_gt_0_iff
tff(fact_1816_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_list_nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(A3)),aa_nat_nat(suc,zero_zero_nat)))
      <=> ! [X3: list_nat] :
            ( member_list_nat(X3,A3)
           => ! [Xa3: list_nat] :
                ( member_list_nat(Xa3,A3)
               => ( X3 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_1817_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(A3)),aa_nat_nat(suc,zero_zero_nat)))
      <=> ! [X3: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X3),A3))
           => ! [Xa3: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(Xa3),A3))
               => ( X3 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_1818_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_nat_nat(suc,zero_zero_nat)))
      <=> ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A3))
           => ! [Xa3: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa3),A3))
               => ( X3 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_1819_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(A3)),aa_nat_nat(suc,zero_zero_nat)))
      <=> ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A3))
           => ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),A3))
               => ( X3 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_1820_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(A3)),aa_nat_nat(suc,zero_zero_nat)))
      <=> ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A3))
           => ! [Xa3: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa3),A3))
               => ( X3 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_1821_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),aa_nat_nat(suc,zero_zero_nat)))
      <=> ! [X3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A3))
           => ! [Xa3: product_prod_nat_nat] :
                ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Xa3),A3))
               => ( X3 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_1822_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite121521170596916366d_enat(A3)),aa_nat_nat(suc,zero_zero_nat)))
      <=> ! [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A3))
           => ! [Xa3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa3),A3))
               => ( X3 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_1823_card__psubset,axiom,
    ! [B2: set_list_nat,A3: set_list_nat] :
      ( finite8100373058378681591st_nat(B2)
     => ( ord_le6045566169113846134st_nat(A3,B2)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_list_nat(A3)),finite_card_list_nat(B2)))
         => ord_le1190675801316882794st_nat(A3,B2) ) ) ) ).

% card_psubset
tff(fact_1824_card__psubset,axiom,
    ! [B2: set_set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B2))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_set_nat(A3)),finite_card_set_nat(B2)))
         => pp(aa_set_set_nat_bool(ord_less_set_set_nat(A3),B2)) ) ) ) ).

% card_psubset
tff(fact_1825_card__psubset,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,B2)))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A3),B2)) ) ) ) ).

% card_psubset
tff(fact_1826_card__psubset,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_complex(A3)),finite_card_complex(B2)))
         => pp(aa_set_complex_bool(ord_less_set_complex(A3),B2)) ) ) ) ).

% card_psubset
tff(fact_1827_card__psubset,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B2)))
         => pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A3),B2)) ) ) ) ).

% card_psubset
tff(fact_1828_card__psubset,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite121521170596916366d_enat(A3)),finite121521170596916366d_enat(B2)))
         => pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(A3),B2)) ) ) ) ).

% card_psubset
tff(fact_1829_card__psubset,axiom,
    ! [B2: set_int,A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_int(A3)),finite_card_int(B2)))
         => pp(aa_set_int_bool(ord_less_set_int(A3),B2)) ) ) ) ).

% card_psubset
tff(fact_1830_finite__enumerate__mono,axiom,
    ! [M: nat,N: nat,S: set_Extended_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),finite121521170596916366d_enat(S)))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,infini7641415182203889163d_enat(S,M)),infini7641415182203889163d_enat(S,N))) ) ) ) ).

% finite_enumerate_mono
tff(fact_1831_finite__enumerate__mono,axiom,
    ! [M: nat,N: nat,S: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( pp(aa_set_nat_bool(finite_finite_nat,S))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_set_nat_nat(finite_card_nat,S)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S),M)),aa_nat_nat(infini8530281810654367211te_nat(S),N))) ) ) ) ).

% finite_enumerate_mono
tff(fact_1832_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1833_Suc__pred_H,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_1834_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) )
       => ~ ! [N2: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ) ).

% int_cases3
tff(fact_1835_not__zle__0__negative,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))) ).

% not_zle_0_negative
tff(fact_1836_negD,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
     => ? [N2: nat] : X = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% negD
tff(fact_1837_negative__zless__0,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),zero_zero_int)) ).

% negative_zless_0
tff(fact_1838_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Vj) = none_nat ).

% vebt_pred.simps(6)
tff(fact_1839_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Vi) = none_nat ).

% vebt_succ.simps(5)
tff(fact_1840_finite__le__enumerate,axiom,
    ! [S: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_set_nat_nat(finite_card_nat,S)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(infini8530281810654367211te_nat(S),N))) ) ) ).

% finite_le_enumerate
tff(fact_1841_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
     => ~ ! [N2: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),E2)) ) ).

% nat_approx_posE
tff(fact_1842_nat__approx__posE,axiom,
    ! [E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ~ ! [N2: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))),E2)) ) ).

% nat_approx_posE
tff(fact_1843_finite__enumerate__step,axiom,
    ! [S: set_Extended_enat,N: nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),finite121521170596916366d_enat(S)))
       => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,infini7641415182203889163d_enat(S,N)),infini7641415182203889163d_enat(S,aa_nat_nat(suc,N)))) ) ) ).

% finite_enumerate_step
tff(fact_1844_finite__enumerate__step,axiom,
    ! [S: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),aa_set_nat_nat(finite_card_nat,S)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S),N)),aa_nat_nat(infini8530281810654367211te_nat(S),aa_nat_nat(suc,N)))) ) ) ).

% finite_enumerate_step
tff(fact_1845_le__divide__eq__1__pos,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),A2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ) ).

% le_divide_eq_1_pos
tff(fact_1846_le__divide__eq__1__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),A2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ) ).

% le_divide_eq_1_pos
tff(fact_1847_le__divide__eq__1__neg,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),A2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2)) ) ) ).

% le_divide_eq_1_neg
tff(fact_1848_le__divide__eq__1__neg,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),A2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2)) ) ) ).

% le_divide_eq_1_neg
tff(fact_1849_divide__le__eq__1__pos,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),A2)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2)) ) ) ).

% divide_le_eq_1_pos
tff(fact_1850_divide__le__eq__1__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),A2)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2)) ) ) ).

% divide_le_eq_1_pos
tff(fact_1851_divide__le__eq__1__neg,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),A2)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ) ).

% divide_le_eq_1_neg
tff(fact_1852_divide__le__eq__1__neg,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),A2)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ) ).

% divide_le_eq_1_neg
tff(fact_1853_zero__less__divide__1__iff,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2)) ) ).

% zero_less_divide_1_iff
tff(fact_1854_zero__less__divide__1__iff,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2)) ) ).

% zero_less_divide_1_iff
tff(fact_1855_less__divide__eq__1__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),A2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) ) ) ).

% less_divide_eq_1_pos
tff(fact_1856_less__divide__eq__1__pos,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),A2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) ) ) ).

% less_divide_eq_1_pos
tff(fact_1857_less__divide__eq__1__neg,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),A2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2)) ) ) ).

% less_divide_eq_1_neg
tff(fact_1858_less__divide__eq__1__neg,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),A2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2)) ) ) ).

% less_divide_eq_1_neg
tff(fact_1859_divide__less__eq__1__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),A2)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2)) ) ) ).

% divide_less_eq_1_pos
tff(fact_1860_divide__less__eq__1__pos,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),A2)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2)) ) ) ).

% divide_less_eq_1_pos
tff(fact_1861_divide__less__eq__1__neg,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),A2)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) ) ) ).

% divide_less_eq_1_neg
tff(fact_1862_divide__less__eq__1__neg,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),A2)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) ) ) ).

% divide_less_eq_1_neg
tff(fact_1863_divide__less__0__1__iff,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A2)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ).

% divide_less_0_1_iff
tff(fact_1864_divide__less__0__1__iff,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ).

% divide_less_0_1_iff
tff(fact_1865_Diff__empty,axiom,
    ! [A3: set_real] : minus_minus_set_real(A3,bot_bot_set_real) = A3 ).

% Diff_empty
tff(fact_1866_Diff__empty,axiom,
    ! [A3: set_bool] : minus_minus_set_bool(A3,bot_bot_set_bool) = A3 ).

% Diff_empty
tff(fact_1867_Diff__empty,axiom,
    ! [A3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),bot_bot_set_int) = A3 ).

% Diff_empty
tff(fact_1868_Diff__empty,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),bot_bot_set_nat) = A3 ).

% Diff_empty
tff(fact_1869_empty__Diff,axiom,
    ! [A3: set_real] : minus_minus_set_real(bot_bot_set_real,A3) = bot_bot_set_real ).

% empty_Diff
tff(fact_1870_empty__Diff,axiom,
    ! [A3: set_bool] : minus_minus_set_bool(bot_bot_set_bool,A3) = bot_bot_set_bool ).

% empty_Diff
tff(fact_1871_empty__Diff,axiom,
    ! [A3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,bot_bot_set_int),A3) = bot_bot_set_int ).

% empty_Diff
tff(fact_1872_empty__Diff,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,bot_bot_set_nat),A3) = bot_bot_set_nat ).

% empty_Diff
tff(fact_1873_Diff__cancel,axiom,
    ! [A3: set_real] : minus_minus_set_real(A3,A3) = bot_bot_set_real ).

% Diff_cancel
tff(fact_1874_Diff__cancel,axiom,
    ! [A3: set_bool] : minus_minus_set_bool(A3,A3) = bot_bot_set_bool ).

% Diff_cancel
tff(fact_1875_Diff__cancel,axiom,
    ! [A3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),A3) = bot_bot_set_int ).

% Diff_cancel
tff(fact_1876_Diff__cancel,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),A3) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_1877_finite__Diff,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2))) ) ).

% finite_Diff
tff(fact_1878_finite__Diff,axiom,
    ! [A3: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A3,B2))) ) ).

% finite_Diff
tff(fact_1879_finite__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2))) ) ).

% finite_Diff
tff(fact_1880_finite__Diff,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,minus_925952699566721837d_enat(A3,B2))) ) ).

% finite_Diff
tff(fact_1881_finite__Diff,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => pp(aa_set_nat_bool(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2))) ) ).

% finite_Diff
tff(fact_1882_finite__Diff2,axiom,
    ! [B2: set_int,A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2)))
      <=> pp(aa_set_int_bool(finite_finite_int,A3)) ) ) ).

% finite_Diff2
tff(fact_1883_finite__Diff2,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A3,B2)))
      <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,A3)) ) ) ).

% finite_Diff2
tff(fact_1884_finite__Diff2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
     => ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2)))
      <=> pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3)) ) ) ).

% finite_Diff2
tff(fact_1885_finite__Diff2,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,minus_925952699566721837d_enat(A3,B2)))
      <=> pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3)) ) ) ).

% finite_Diff2
tff(fact_1886_finite__Diff2,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)))
      <=> pp(aa_set_nat_bool(finite_finite_nat,A3)) ) ) ).

% finite_Diff2
tff(fact_1887_Compl__anti__mono,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(B2)),uminus1532241313380277803et_int(A3))) ) ).

% Compl_anti_mono
tff(fact_1888_Compl__subset__Compl__iff,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(A3)),uminus1532241313380277803et_int(B2)))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(B2),A3)) ) ).

% Compl_subset_Compl_iff
tff(fact_1889_divide__eq__0__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3) = zero_zero_rat )
    <=> ( ( A2 = zero_zero_rat )
        | ( B3 = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_1890_divide__eq__0__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3) = zero_zero_real )
    <=> ( ( A2 = zero_zero_real )
        | ( B3 = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_1891_divide__eq__0__iff,axiom,
    ! [A2: complex,B3: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A2),B3) = zero_zero_complex )
    <=> ( ( A2 = zero_zero_complex )
        | ( B3 = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
tff(fact_1892_divide__cancel__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B3) )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = B3 ) ) ) ).

% divide_cancel_left
tff(fact_1893_divide__cancel__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B3) )
    <=> ( ( C = zero_zero_real )
        | ( A2 = B3 ) ) ) ).

% divide_cancel_left
tff(fact_1894_divide__cancel__left,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(C),A2) = aa_complex_complex(divide1717551699836669952omplex(C),B3) )
    <=> ( ( C = zero_zero_complex )
        | ( A2 = B3 ) ) ) ).

% divide_cancel_left
tff(fact_1895_divide__cancel__right,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),C) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = B3 ) ) ) ).

% divide_cancel_right
tff(fact_1896_divide__cancel__right,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C) )
    <=> ( ( C = zero_zero_real )
        | ( A2 = B3 ) ) ) ).

% divide_cancel_right
tff(fact_1897_divide__cancel__right,axiom,
    ! [A2: complex,C: complex,B3: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A2),C) = aa_complex_complex(divide1717551699836669952omplex(B3),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A2 = B3 ) ) ) ).

% divide_cancel_right
tff(fact_1898_division__ring__divide__zero,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_1899_division__ring__divide__zero,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_1900_division__ring__divide__zero,axiom,
    ! [A2: complex] : aa_complex_complex(divide1717551699836669952omplex(A2),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_1901_Diff__eq__empty__iff,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ( minus_minus_set_real(A3,B2) = bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2)) ) ).

% Diff_eq_empty_iff
tff(fact_1902_Diff__eq__empty__iff,axiom,
    ! [A3: set_bool,B2: set_bool] :
      ( ( minus_minus_set_bool(A3,B2) = bot_bot_set_bool )
    <=> pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),B2)) ) ).

% Diff_eq_empty_iff
tff(fact_1903_Diff__eq__empty__iff,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2)) ) ).

% Diff_eq_empty_iff
tff(fact_1904_Diff__eq__empty__iff,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2)) ) ).

% Diff_eq_empty_iff
tff(fact_1905_divide__eq__1__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3) = one_one_rat )
    <=> ( ( B3 != zero_zero_rat )
        & ( A2 = B3 ) ) ) ).

% divide_eq_1_iff
tff(fact_1906_divide__eq__1__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3) = one_one_real )
    <=> ( ( B3 != zero_zero_real )
        & ( A2 = B3 ) ) ) ).

% divide_eq_1_iff
tff(fact_1907_divide__eq__1__iff,axiom,
    ! [A2: complex,B3: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A2),B3) = one_one_complex )
    <=> ( ( B3 != zero_zero_complex )
        & ( A2 = B3 ) ) ) ).

% divide_eq_1_iff
tff(fact_1908_one__eq__divide__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( one_one_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3) )
    <=> ( ( B3 != zero_zero_rat )
        & ( A2 = B3 ) ) ) ).

% one_eq_divide_iff
tff(fact_1909_one__eq__divide__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3) )
    <=> ( ( B3 != zero_zero_real )
        & ( A2 = B3 ) ) ) ).

% one_eq_divide_iff
tff(fact_1910_one__eq__divide__iff,axiom,
    ! [A2: complex,B3: complex] :
      ( ( one_one_complex = aa_complex_complex(divide1717551699836669952omplex(A2),B3) )
    <=> ( ( B3 != zero_zero_complex )
        & ( A2 = B3 ) ) ) ).

% one_eq_divide_iff
tff(fact_1911_divide__self,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),A2) = one_one_rat ) ) ).

% divide_self
tff(fact_1912_divide__self,axiom,
    ! [A2: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),A2) = one_one_real ) ) ).

% divide_self
tff(fact_1913_divide__self,axiom,
    ! [A2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A2),A2) = one_one_complex ) ) ).

% divide_self
tff(fact_1914_divide__self__if,axiom,
    ! [A2: rat] :
      ( ( ( A2 = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),A2) = zero_zero_rat ) )
      & ( ( A2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),A2) = one_one_rat ) ) ) ).

% divide_self_if
tff(fact_1915_divide__self__if,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),A2) = zero_zero_real ) )
      & ( ( A2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),A2) = one_one_real ) ) ) ).

% divide_self_if
tff(fact_1916_divide__self__if,axiom,
    ! [A2: complex] :
      ( ( ( A2 = zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(A2),A2) = zero_zero_complex ) )
      & ( ( A2 != zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(A2),A2) = one_one_complex ) ) ) ).

% divide_self_if
tff(fact_1917_divide__eq__eq__1,axiom,
    ! [B3: rat,A2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),A2) = one_one_rat )
    <=> ( ( A2 != zero_zero_rat )
        & ( A2 = B3 ) ) ) ).

% divide_eq_eq_1
tff(fact_1918_divide__eq__eq__1,axiom,
    ! [B3: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),A2) = one_one_real )
    <=> ( ( A2 != zero_zero_real )
        & ( A2 = B3 ) ) ) ).

% divide_eq_eq_1
tff(fact_1919_eq__divide__eq__1,axiom,
    ! [B3: rat,A2: rat] :
      ( ( one_one_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),A2) )
    <=> ( ( A2 != zero_zero_rat )
        & ( A2 = B3 ) ) ) ).

% eq_divide_eq_1
tff(fact_1920_eq__divide__eq__1,axiom,
    ! [B3: real,A2: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),A2) )
    <=> ( ( A2 != zero_zero_real )
        & ( A2 = B3 ) ) ) ).

% eq_divide_eq_1
tff(fact_1921_one__divide__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_1922_one__divide__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_1923_zero__eq__1__divide__iff,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A2) )
    <=> ( A2 = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_1924_zero__eq__1__divide__iff,axiom,
    ! [A2: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2) )
    <=> ( A2 = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_1925_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ).

% zle_diff1_eq
tff(fact_1926_zero__le__divide__1__iff,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2)) ) ).

% zero_le_divide_1_iff
tff(fact_1927_zero__le__divide__1__iff,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2)) ) ).

% zero_le_divide_1_iff
tff(fact_1928_divide__le__0__1__iff,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) ) ).

% divide_le_0_1_iff
tff(fact_1929_divide__le__0__1__iff,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A2)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) ) ).

% divide_le_0_1_iff
tff(fact_1930_Diff__infinite__finite,axiom,
    ! [T2: set_int,S: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( ~ pp(aa_set_int_bool(finite_finite_int,S))
       => ~ pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),T2))) ) ) ).

% Diff_infinite_finite
tff(fact_1931_Diff__infinite__finite,axiom,
    ! [T2: set_complex,S: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(S,T2))) ) ) ).

% Diff_infinite_finite
tff(fact_1932_Diff__infinite__finite,axiom,
    ! [T2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,T2))
     => ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S))
       => ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,S),T2))) ) ) ).

% Diff_infinite_finite
tff(fact_1933_Diff__infinite__finite,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T2))
     => ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
       => ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,minus_925952699566721837d_enat(S,T2))) ) ) ).

% Diff_infinite_finite
tff(fact_1934_Diff__infinite__finite,axiom,
    ! [T2: set_nat,S: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
       => ~ pp(aa_set_nat_bool(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),T2))) ) ) ).

% Diff_infinite_finite
tff(fact_1935_Diff__mono,axiom,
    ! [A3: set_nat,C2: set_nat,D4: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),C2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,D4),B2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),D4))) ) ) ).

% Diff_mono
tff(fact_1936_Diff__mono,axiom,
    ! [A3: set_int,C2: set_int,D4: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),C2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(D4),B2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2)),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,C2),D4))) ) ) ).

% Diff_mono
tff(fact_1937_Diff__subset,axiom,
    ! [A3: set_nat,B2: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)),A3)) ).

% Diff_subset
tff(fact_1938_Diff__subset,axiom,
    ! [A3: set_int,B2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2)),A3)) ).

% Diff_subset
tff(fact_1939_double__diff,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),C2))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A3)) = A3 ) ) ) ).

% double_diff
tff(fact_1940_double__diff,axiom,
    ! [A3: set_int,B2: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C2))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,C2),A3)) = A3 ) ) ) ).

% double_diff
tff(fact_1941_psubset__imp__ex__mem,axiom,
    ! [A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A3),B2))
     => ? [B: real] : pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A3))) ) ).

% psubset_imp_ex_mem
tff(fact_1942_psubset__imp__ex__mem,axiom,
    ! [A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_set_bool(A3),B2))
     => ? [B: bool] : pp(aa_set_bool_bool(member_bool(B),minus_minus_set_bool(B2,A3))) ) ).

% psubset_imp_ex_mem
tff(fact_1943_psubset__imp__ex__mem,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A3),B2))
     => ? [B: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(B),minus_2163939370556025621et_nat(B2,A3))) ) ).

% psubset_imp_ex_mem
tff(fact_1944_psubset__imp__ex__mem,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A3),B2))
     => ? [B: int] : pp(aa_set_int_bool(member_int(B),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B2),A3))) ) ).

% psubset_imp_ex_mem
tff(fact_1945_psubset__imp__ex__mem,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A3),B2))
     => ? [B: nat] : pp(aa_set_nat_bool(member_nat(B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A3))) ) ).

% psubset_imp_ex_mem
tff(fact_1946_subset__Compl__self__eq,axiom,
    ! [A3: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),uminus612125837232591019t_real(A3)))
    <=> ( A3 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
tff(fact_1947_subset__Compl__self__eq,axiom,
    ! [A3: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),uminus5166143097222041326t_bool(A3)))
    <=> ( A3 = bot_bot_set_bool ) ) ).

% subset_Compl_self_eq
tff(fact_1948_subset__Compl__self__eq,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),uminus5710092332889474511et_nat(A3)))
    <=> ( A3 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_1949_subset__Compl__self__eq,axiom,
    ! [A3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),uminus1532241313380277803et_int(A3)))
    <=> ( A3 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_1950_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),K))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_le_induct
tff(fact_1951_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K))
     => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),K))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_less_induct
tff(fact_1952_card__less__sym__Diff,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( finite8100373058378681591st_nat(B2)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_list_nat(A3)),finite_card_list_nat(B2)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A3,B2))),finite_card_list_nat(minus_7954133019191499631st_nat(B2,A3)))) ) ) ) ).

% card_less_sym_Diff
tff(fact_1953_card__less__sym__Diff,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_set_nat(A3)),finite_card_set_nat(B2)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A3,B2))),finite_card_set_nat(minus_2163939370556025621et_nat(B2,A3)))) ) ) ) ).

% card_less_sym_Diff
tff(fact_1954_card__less__sym__Diff,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_int(A3)),finite_card_int(B2)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2))),finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B2),A3)))) ) ) ) ).

% card_less_sym_Diff
tff(fact_1955_card__less__sym__Diff,axiom,
    ! [A3: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_complex(A3)),finite_card_complex(B2)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_complex(minus_811609699411566653omplex(A3,B2))),finite_card_complex(minus_811609699411566653omplex(B2,A3)))) ) ) ) ).

% card_less_sym_Diff
tff(fact_1956_card__less__sym__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B2)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2))),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),A3)))) ) ) ) ).

% card_less_sym_Diff
tff(fact_1957_card__less__sym__Diff,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite121521170596916366d_enat(A3)),finite121521170596916366d_enat(B2)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite121521170596916366d_enat(minus_925952699566721837d_enat(A3,B2))),finite121521170596916366d_enat(minus_925952699566721837d_enat(B2,A3)))) ) ) ) ).

% card_less_sym_Diff
tff(fact_1958_card__less__sym__Diff,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,B2)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2))),aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A3)))) ) ) ) ).

% card_less_sym_Diff
tff(fact_1959_card__le__sym__Diff,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( finite8100373058378681591st_nat(B2)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(A3)),finite_card_list_nat(B2)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A3,B2))),finite_card_list_nat(minus_7954133019191499631st_nat(B2,A3)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_1960_card__le__sym__Diff,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(A3)),finite_card_set_nat(B2)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A3,B2))),finite_card_set_nat(minus_2163939370556025621et_nat(B2,A3)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_1961_card__le__sym__Diff,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(A3)),finite_card_int(B2)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2))),finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B2),A3)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_1962_card__le__sym__Diff,axiom,
    ! [A3: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(A3)),finite_card_complex(B2)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(minus_811609699411566653omplex(A3,B2))),finite_card_complex(minus_811609699411566653omplex(B2,A3)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_1963_card__le__sym__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B2)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2))),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),A3)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_1964_card__le__sym__Diff,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite121521170596916366d_enat(A3)),finite121521170596916366d_enat(B2)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite121521170596916366d_enat(minus_925952699566721837d_enat(A3,B2))),finite121521170596916366d_enat(minus_925952699566721837d_enat(B2,A3)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_1965_card__le__sym__Diff,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,B2)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2))),aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A3)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_1966_linordered__field__no__ub,axiom,
    ! [X4: real] :
    ? [X_12: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),X_12)) ).

% linordered_field_no_ub
tff(fact_1967_linordered__field__no__ub,axiom,
    ! [X4: rat] :
    ? [X_12: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),X_12)) ).

% linordered_field_no_ub
tff(fact_1968_linordered__field__no__lb,axiom,
    ! [X4: real] :
    ? [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X4)) ).

% linordered_field_no_lb
tff(fact_1969_linordered__field__no__lb,axiom,
    ! [X4: rat] :
    ? [Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X4)) ).

% linordered_field_no_lb
tff(fact_1970_card__Diff__subset,axiom,
    ! [B2: set_list_nat,A3: set_list_nat] :
      ( finite8100373058378681591st_nat(B2)
     => ( ord_le6045566169113846134st_nat(B2,A3)
       => ( finite_card_list_nat(minus_7954133019191499631st_nat(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A3)),finite_card_list_nat(B2)) ) ) ) ).

% card_Diff_subset
tff(fact_1971_card__Diff__subset,axiom,
    ! [B2: set_set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B2))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(B2),A3))
       => ( finite_card_set_nat(minus_2163939370556025621et_nat(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A3)),finite_card_set_nat(B2)) ) ) ) ).

% card_Diff_subset
tff(fact_1972_card__Diff__subset,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A3))
       => ( finite_card_complex(minus_811609699411566653omplex(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A3)),finite_card_complex(B2)) ) ) ) ).

% card_Diff_subset
tff(fact_1973_card__Diff__subset,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(B2),A3))
       => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B2)) ) ) ) ).

% card_Diff_subset
tff(fact_1974_card__Diff__subset,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),A3))
       => ( finite121521170596916366d_enat(minus_925952699566721837d_enat(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite121521170596916366d_enat(A3)),finite121521170596916366d_enat(B2)) ) ) ) ).

% card_Diff_subset
tff(fact_1975_card__Diff__subset,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A3))
       => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,B2)) ) ) ) ).

% card_Diff_subset
tff(fact_1976_card__Diff__subset,axiom,
    ! [B2: set_int,A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A3))
       => ( finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A3)),finite_card_int(B2)) ) ) ) ).

% card_Diff_subset
tff(fact_1977_diff__card__le__card__Diff,axiom,
    ! [B2: set_list_nat,A3: set_list_nat] :
      ( finite8100373058378681591st_nat(B2)
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A3)),finite_card_list_nat(B2))),finite_card_list_nat(minus_7954133019191499631st_nat(A3,B2)))) ) ).

% diff_card_le_card_Diff
tff(fact_1978_diff__card__le__card__Diff,axiom,
    ! [B2: set_set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A3)),finite_card_set_nat(B2))),finite_card_set_nat(minus_2163939370556025621et_nat(A3,B2)))) ) ).

% diff_card_le_card_Diff
tff(fact_1979_diff__card__le__card__Diff,axiom,
    ! [B2: set_int,A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A3)),finite_card_int(B2))),finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2)))) ) ).

% diff_card_le_card_Diff
tff(fact_1980_diff__card__le__card__Diff,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A3)),finite_card_complex(B2))),finite_card_complex(minus_811609699411566653omplex(A3,B2)))) ) ).

% diff_card_le_card_Diff
tff(fact_1981_diff__card__le__card__Diff,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B2))),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2)))) ) ).

% diff_card_le_card_Diff
tff(fact_1982_diff__card__le__card__Diff,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite121521170596916366d_enat(A3)),finite121521170596916366d_enat(B2))),finite121521170596916366d_enat(minus_925952699566721837d_enat(A3,B2)))) ) ).

% diff_card_le_card_Diff
tff(fact_1983_diff__card__le__card__Diff,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,B2))),aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)))) ) ).

% diff_card_le_card_Diff
tff(fact_1984_int__ops_I6_J,axiom,
    ! [A2: nat,B3: nat] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B3)))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B3)) = zero_zero_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B3)))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B3)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B3)) ) ) ) ).

% int_ops(6)
tff(fact_1985_VEBT_Osize_I4_J,axiom,
    ! [X21: bool,X222: bool] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf(X21,X222)) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_1986_zdiff__int__split,axiom,
    ! [P: fun_int_bool,X: nat,Y: nat] :
      ( pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Y))))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
         => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)))) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
         => pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% zdiff_int_split
tff(fact_1987_divide__right__mono__neg,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C))) ) ) ).

% divide_right_mono_neg
tff(fact_1988_divide__right__mono__neg,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),C))) ) ) ).

% divide_right_mono_neg
tff(fact_1989_divide__nonpos__nonpos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_1990_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_1991_divide__nonpos__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_nonneg
tff(fact_1992_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_nonneg
tff(fact_1993_divide__nonneg__nonpos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_nonpos
tff(fact_1994_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_nonpos
tff(fact_1995_divide__nonneg__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_1996_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_1997_zero__le__divide__iff,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),zero_zero_real)) ) ) ) ).

% zero_le_divide_iff
tff(fact_1998_zero__le__divide__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),zero_zero_rat)) ) ) ) ).

% zero_le_divide_iff
tff(fact_1999_divide__right__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C))) ) ) ).

% divide_right_mono
tff(fact_2000_divide__right__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C))) ) ) ).

% divide_right_mono
tff(fact_2001_divide__le__0__iff,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3)) ) ) ) ).

% divide_le_0_iff
tff(fact_2002_divide__le__0__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3)) ) ) ) ).

% divide_le_0_iff
tff(fact_2003_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y))) ) ) ).

% divide_neg_neg
tff(fact_2004_divide__neg__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_neg_neg
tff(fact_2005_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat)) ) ) ).

% divide_neg_pos
tff(fact_2006_divide__neg__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_neg_pos
tff(fact_2007_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat)) ) ) ).

% divide_pos_neg
tff(fact_2008_divide__pos__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_pos_neg
tff(fact_2009_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y))) ) ) ).

% divide_pos_pos
tff(fact_2010_divide__pos__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_pos_pos
tff(fact_2011_divide__less__0__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3)) ) ) ) ).

% divide_less_0_iff
tff(fact_2012_divide__less__0__iff,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3)) ) ) ) ).

% divide_less_0_iff
tff(fact_2013_divide__less__cancel,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2)) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_2014_divide__less__cancel,axiom,
    ! [A2: real,C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2)) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_2015_zero__less__divide__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),zero_zero_rat)) ) ) ) ).

% zero_less_divide_iff
tff(fact_2016_zero__less__divide__iff,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),zero_zero_real)) ) ) ) ).

% zero_less_divide_iff
tff(fact_2017_divide__strict__right__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C))) ) ) ).

% divide_strict_right_mono
tff(fact_2018_divide__strict__right__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C))) ) ) ).

% divide_strict_right_mono
tff(fact_2019_divide__strict__right__mono__neg,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2020_divide__strict__right__mono__neg,axiom,
    ! [B3: real,A2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2021_right__inverse__eq,axiom,
    ! [B3: rat,A2: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3) = one_one_rat )
      <=> ( A2 = B3 ) ) ) ).

% right_inverse_eq
tff(fact_2022_right__inverse__eq,axiom,
    ! [B3: real,A2: real] :
      ( ( B3 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3) = one_one_real )
      <=> ( A2 = B3 ) ) ) ).

% right_inverse_eq
tff(fact_2023_right__inverse__eq,axiom,
    ! [B3: complex,A2: complex] :
      ( ( B3 != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A2),B3) = one_one_complex )
      <=> ( A2 = B3 ) ) ) ).

% right_inverse_eq
tff(fact_2024_nonzero__minus__divide__divide,axiom,
    ! [B3: real,A2: real] :
      ( ( B3 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(uminus_uminus_real,B3)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3) ) ) ).

% nonzero_minus_divide_divide
tff(fact_2025_nonzero__minus__divide__divide,axiom,
    ! [B3: rat,A2: rat] :
      ( ( B3 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(uminus_uminus_rat,B3)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3) ) ) ).

% nonzero_minus_divide_divide
tff(fact_2026_nonzero__minus__divide__divide,axiom,
    ! [B3: complex,A2: complex] :
      ( ( B3 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(uminus1482373934393186551omplex,B3)) = aa_complex_complex(divide1717551699836669952omplex(A2),B3) ) ) ).

% nonzero_minus_divide_divide
tff(fact_2027_nonzero__minus__divide__right,axiom,
    ! [B3: real,A2: real] :
      ( ( B3 != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(uminus_uminus_real,B3)) ) ) ).

% nonzero_minus_divide_right
tff(fact_2028_nonzero__minus__divide__right,axiom,
    ! [B3: rat,A2: rat] :
      ( ( B3 != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),aa_rat_rat(uminus_uminus_rat,B3)) ) ) ).

% nonzero_minus_divide_right
tff(fact_2029_nonzero__minus__divide__right,axiom,
    ! [B3: complex,A2: complex] :
      ( ( B3 != zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A2),B3)) = aa_complex_complex(divide1717551699836669952omplex(A2),aa_complex_complex(uminus1482373934393186551omplex,B3)) ) ) ).

% nonzero_minus_divide_right
tff(fact_2030_frac__le,axiom,
    ! [Y: real,X: real,W2: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W2),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2))) ) ) ) ) ).

% frac_le
tff(fact_2031_frac__le,axiom,
    ! [Y: rat,X: rat,W2: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W2),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W2))) ) ) ) ) ).

% frac_le
tff(fact_2032_frac__less,axiom,
    ! [X: real,Y: real,W2: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W2),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2))) ) ) ) ) ).

% frac_less
tff(fact_2033_frac__less,axiom,
    ! [X: rat,Y: rat,W2: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W2),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W2))) ) ) ) ) ).

% frac_less
tff(fact_2034_frac__less2,axiom,
    ! [X: real,Y: real,W2: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W2),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2))) ) ) ) ) ).

% frac_less2
tff(fact_2035_frac__less2,axiom,
    ! [X: rat,Y: rat,W2: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W2),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W2))) ) ) ) ) ).

% frac_less2
tff(fact_2036_divide__le__cancel,axiom,
    ! [A2: real,C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2)) ) ) ) ).

% divide_le_cancel
tff(fact_2037_divide__le__cancel,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2)) ) ) ) ).

% divide_le_cancel
tff(fact_2038_divide__nonneg__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_neg
tff(fact_2039_divide__nonneg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_neg
tff(fact_2040_divide__nonneg__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonneg_pos
tff(fact_2041_divide__nonneg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y))) ) ) ).

% divide_nonneg_pos
tff(fact_2042_divide__nonpos__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonpos_neg
tff(fact_2043_divide__nonpos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y))) ) ) ).

% divide_nonpos_neg
tff(fact_2044_divide__nonpos__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_pos
tff(fact_2045_divide__nonpos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_pos
tff(fact_2046_divide__less__eq__1,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),A2)),one_one_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) )
        | ( A2 = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_2047_divide__less__eq__1,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),A2)),one_one_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) )
        | ( A2 = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_2048_less__divide__eq__1,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),A2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2)) ) ) ) ).

% less_divide_eq_1
tff(fact_2049_less__divide__eq__1,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),A2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2)) ) ) ) ).

% less_divide_eq_1
tff(fact_2050_divide__eq__minus__1__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B3 != zero_zero_real )
        & ( A2 = aa_real_real(uminus_uminus_real,B3) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_2051_divide__eq__minus__1__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( ( B3 != zero_zero_rat )
        & ( A2 = aa_rat_rat(uminus_uminus_rat,B3) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_2052_divide__eq__minus__1__iff,axiom,
    ! [A2: complex,B3: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A2),B3) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( B3 != zero_zero_complex )
        & ( A2 = aa_complex_complex(uminus1482373934393186551omplex,B3) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_2053_divide__le__eq__1,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),A2)),one_one_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) )
        | ( A2 = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_2054_divide__le__eq__1,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),A2)),one_one_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) )
        | ( A2 = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_2055_le__divide__eq__1,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),A2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2)) ) ) ) ).

% le_divide_eq_1
tff(fact_2056_le__divide__eq__1,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),A2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2)) ) ) ) ).

% le_divide_eq_1
tff(fact_2057_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_2058_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_2059_div__eq__minus1,axiom,
    ! [B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),B3) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_2060_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Ma),X))
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = none_nat ) ) ) ).

% geqmaxNone
tff(fact_2061_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_2062_div__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) ) ).

% div_less
tff(fact_2063_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_2064_compl__le__compl__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(X)),uminus1532241313380277803et_int(Y)))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(Y),X)) ) ).

% compl_le_compl_iff
tff(fact_2065_bits__div__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A2) = zero_zero_int ).

% bits_div_0
tff(fact_2066_bits__div__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A2) = zero_zero_nat ).

% bits_div_0
tff(fact_2067_DiffI,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),A3))
     => ( ~ pp(aa_set_real_bool(member_real(C),B2))
       => pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A3,B2))) ) ) ).

% DiffI
tff(fact_2068_DiffI,axiom,
    ! [C: bool,A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(C),A3))
     => ( ~ pp(aa_set_bool_bool(member_bool(C),B2))
       => pp(aa_set_bool_bool(member_bool(C),minus_minus_set_bool(A3,B2))) ) ) ).

% DiffI
tff(fact_2069_DiffI,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),A3))
     => ( ~ pp(aa_set_set_nat_bool(member_set_nat(C),B2))
       => pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A3,B2))) ) ) ).

% DiffI
tff(fact_2070_DiffI,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),A3))
     => ( ~ pp(aa_set_int_bool(member_int(C),B2))
       => pp(aa_set_int_bool(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2))) ) ) ).

% DiffI
tff(fact_2071_DiffI,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),A3))
     => ( ~ pp(aa_set_nat_bool(member_nat(C),B2))
       => pp(aa_set_nat_bool(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2))) ) ) ).

% DiffI
tff(fact_2072_Diff__iff,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A3,B2)))
    <=> ( pp(aa_set_real_bool(member_real(C),A3))
        & ~ pp(aa_set_real_bool(member_real(C),B2)) ) ) ).

% Diff_iff
tff(fact_2073_Diff__iff,axiom,
    ! [C: bool,A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(C),minus_minus_set_bool(A3,B2)))
    <=> ( pp(aa_set_bool_bool(member_bool(C),A3))
        & ~ pp(aa_set_bool_bool(member_bool(C),B2)) ) ) ).

% Diff_iff
tff(fact_2074_Diff__iff,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A3,B2)))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(C),A3))
        & ~ pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ) ).

% Diff_iff
tff(fact_2075_Diff__iff,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2)))
    <=> ( pp(aa_set_int_bool(member_int(C),A3))
        & ~ pp(aa_set_int_bool(member_int(C),B2)) ) ) ).

% Diff_iff
tff(fact_2076_Diff__iff,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)))
    <=> ( pp(aa_set_nat_bool(member_nat(C),A3))
        & ~ pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).

% Diff_iff
tff(fact_2077_Diff__idemp,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2) ).

% Diff_idemp
tff(fact_2078_ComplI,axiom,
    ! [C: real,A3: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(C),A3))
     => pp(aa_set_real_bool(member_real(C),uminus612125837232591019t_real(A3))) ) ).

% ComplI
tff(fact_2079_ComplI,axiom,
    ! [C: bool,A3: set_bool] :
      ( ~ pp(aa_set_bool_bool(member_bool(C),A3))
     => pp(aa_set_bool_bool(member_bool(C),uminus5166143097222041326t_bool(A3))) ) ).

% ComplI
tff(fact_2080_ComplI,axiom,
    ! [C: set_nat,A3: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(C),A3))
     => pp(aa_set_set_nat_bool(member_set_nat(C),uminus613421341184616069et_nat(A3))) ) ).

% ComplI
tff(fact_2081_ComplI,axiom,
    ! [C: nat,A3: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(C),A3))
     => pp(aa_set_nat_bool(member_nat(C),uminus5710092332889474511et_nat(A3))) ) ).

% ComplI
tff(fact_2082_ComplI,axiom,
    ! [C: int,A3: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(C),A3))
     => pp(aa_set_int_bool(member_int(C),uminus1532241313380277803et_int(A3))) ) ).

% ComplI
tff(fact_2083_Compl__iff,axiom,
    ! [C: real,A3: set_real] :
      ( pp(aa_set_real_bool(member_real(C),uminus612125837232591019t_real(A3)))
    <=> ~ pp(aa_set_real_bool(member_real(C),A3)) ) ).

% Compl_iff
tff(fact_2084_Compl__iff,axiom,
    ! [C: bool,A3: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(C),uminus5166143097222041326t_bool(A3)))
    <=> ~ pp(aa_set_bool_bool(member_bool(C),A3)) ) ).

% Compl_iff
tff(fact_2085_Compl__iff,axiom,
    ! [C: set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),uminus613421341184616069et_nat(A3)))
    <=> ~ pp(aa_set_set_nat_bool(member_set_nat(C),A3)) ) ).

% Compl_iff
tff(fact_2086_Compl__iff,axiom,
    ! [C: nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),uminus5710092332889474511et_nat(A3)))
    <=> ~ pp(aa_set_nat_bool(member_nat(C),A3)) ) ).

% Compl_iff
tff(fact_2087_Compl__iff,axiom,
    ! [C: int,A3: set_int] :
      ( pp(aa_set_int_bool(member_int(C),uminus1532241313380277803et_int(A3)))
    <=> ~ pp(aa_set_int_bool(member_int(C),A3)) ) ).

% Compl_iff
tff(fact_2088_bits__div__by__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_2089_bits__div__by__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),zero_zero_nat) = zero_zero_nat ).

% bits_div_by_0
tff(fact_2090_real__of__nat__div3,axiom,
    ! [N: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X)))),one_one_real)) ).

% real_of_nat_div3
tff(fact_2091_DiffE,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A3,B2)))
     => ~ ( pp(aa_set_real_bool(member_real(C),A3))
         => pp(aa_set_real_bool(member_real(C),B2)) ) ) ).

% DiffE
tff(fact_2092_DiffE,axiom,
    ! [C: bool,A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(C),minus_minus_set_bool(A3,B2)))
     => ~ ( pp(aa_set_bool_bool(member_bool(C),A3))
         => pp(aa_set_bool_bool(member_bool(C),B2)) ) ) ).

% DiffE
tff(fact_2093_DiffE,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A3,B2)))
     => ~ ( pp(aa_set_set_nat_bool(member_set_nat(C),A3))
         => pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ) ).

% DiffE
tff(fact_2094_DiffE,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2)))
     => ~ ( pp(aa_set_int_bool(member_int(C),A3))
         => pp(aa_set_int_bool(member_int(C),B2)) ) ) ).

% DiffE
tff(fact_2095_DiffE,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)))
     => ~ ( pp(aa_set_nat_bool(member_nat(C),A3))
         => pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).

% DiffE
tff(fact_2096_ComplD,axiom,
    ! [C: real,A3: set_real] :
      ( pp(aa_set_real_bool(member_real(C),uminus612125837232591019t_real(A3)))
     => ~ pp(aa_set_real_bool(member_real(C),A3)) ) ).

% ComplD
tff(fact_2097_ComplD,axiom,
    ! [C: bool,A3: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(C),uminus5166143097222041326t_bool(A3)))
     => ~ pp(aa_set_bool_bool(member_bool(C),A3)) ) ).

% ComplD
tff(fact_2098_ComplD,axiom,
    ! [C: set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),uminus613421341184616069et_nat(A3)))
     => ~ pp(aa_set_set_nat_bool(member_set_nat(C),A3)) ) ).

% ComplD
tff(fact_2099_ComplD,axiom,
    ! [C: nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),uminus5710092332889474511et_nat(A3)))
     => ~ pp(aa_set_nat_bool(member_nat(C),A3)) ) ).

% ComplD
tff(fact_2100_ComplD,axiom,
    ! [C: int,A3: set_int] :
      ( pp(aa_set_int_bool(member_int(C),uminus1532241313380277803et_int(A3)))
     => ~ pp(aa_set_int_bool(member_int(C),A3)) ) ).

% ComplD
tff(fact_2101_DiffD1,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A3,B2)))
     => pp(aa_set_real_bool(member_real(C),A3)) ) ).

% DiffD1
tff(fact_2102_DiffD1,axiom,
    ! [C: bool,A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(C),minus_minus_set_bool(A3,B2)))
     => pp(aa_set_bool_bool(member_bool(C),A3)) ) ).

% DiffD1
tff(fact_2103_DiffD1,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A3,B2)))
     => pp(aa_set_set_nat_bool(member_set_nat(C),A3)) ) ).

% DiffD1
tff(fact_2104_DiffD1,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2)))
     => pp(aa_set_int_bool(member_int(C),A3)) ) ).

% DiffD1
tff(fact_2105_DiffD1,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)))
     => pp(aa_set_nat_bool(member_nat(C),A3)) ) ).

% DiffD1
tff(fact_2106_DiffD2,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A3,B2)))
     => ~ pp(aa_set_real_bool(member_real(C),B2)) ) ).

% DiffD2
tff(fact_2107_DiffD2,axiom,
    ! [C: bool,A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(C),minus_minus_set_bool(A3,B2)))
     => ~ pp(aa_set_bool_bool(member_bool(C),B2)) ) ).

% DiffD2
tff(fact_2108_DiffD2,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A3,B2)))
     => ~ pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ).

% DiffD2
tff(fact_2109_DiffD2,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2)))
     => ~ pp(aa_set_int_bool(member_int(C),B2)) ) ).

% DiffD2
tff(fact_2110_DiffD2,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)))
     => ~ pp(aa_set_nat_bool(member_nat(C),B2)) ) ).

% DiffD2
tff(fact_2111_real__of__nat__div4,axiom,
    ! [N: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X)))) ).

% real_of_nat_div4
tff(fact_2112_real__of__nat__div2,axiom,
    ! [N: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X))))) ).

% real_of_nat_div2
tff(fact_2113_subrelI,axiom,
    ! [R2: set_Pr6588086440996610945on_nat,S2: set_Pr6588086440996610945on_nat] :
      ( ! [X5: option_nat,Y3: option_nat] :
          ( member4117937158525611210on_nat(produc5098337634421038937on_nat(X5,Y3),R2)
         => member4117937158525611210on_nat(produc5098337634421038937on_nat(X5,Y3),S2) )
     => ord_le6406482658798684961on_nat(R2,S2) ) ).

% subrelI
tff(fact_2114_subrelI,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,S2: set_Pr7459493094073627847at_nat] :
      ( ! [X5: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( member1466754251312161552at_nat(produc9060074326276436823at_nat(X5,Y3),R2)
         => member1466754251312161552at_nat(produc9060074326276436823at_nat(X5,Y3),S2) )
     => ord_le5997549366648089703at_nat(R2,S2) ) ).

% subrelI
tff(fact_2115_subrelI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ! [X5: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Y3),R2)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Y3),S2) )
     => ord_le1268244103169919719at_nat(R2,S2) ) ).

% subrelI
tff(fact_2116_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y3)),R2))
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y3)),S2)) )
     => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(R2),S2)) ) ).

% subrelI
tff(fact_2117_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ! [X5: int,Y3: int] :
          ( member5262025264175285858nt_int(product_Pair_int_int(X5,Y3),R2)
         => member5262025264175285858nt_int(product_Pair_int_int(X5,Y3),S2) )
     => ord_le2843351958646193337nt_int(R2,S2) ) ).

% subrelI
tff(fact_2118_div__le__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M)) ).

% div_le_dividend
tff(fact_2119_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),K))) ) ).

% div_le_mono
tff(fact_2120_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst,Smry),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst,Smry) ).

% vebt_delete.simps(5)
tff(fact_2121_vebt__mint_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A: bool,B: bool] : X != vEBT_Leaf(A,B)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2) ) ) ).

% vebt_mint.cases
tff(fact_2122_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy,Uz)) = some_nat(Mi) ).

% vebt_mint.simps(3)
tff(fact_2123_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va,Vb)),X))
    <=> ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_2124_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy,Uz)) = some_nat(Ma) ).

% vebt_maxt.simps(3)
tff(fact_2125_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm) ).

% vebt_delete.simps(6)
tff(fact_2126_compl__le__swap2,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),X))
     => pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(X)),Y)) ) ).

% compl_le_swap2
tff(fact_2127_compl__le__swap1,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),uminus1532241313380277803et_int(X)))
     => pp(aa_set_int_bool(ord_less_eq_set_int(X),uminus1532241313380277803et_int(Y))) ) ).

% compl_le_swap1
tff(fact_2128_compl__mono,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),uminus1532241313380277803et_int(X))) ) ).

% compl_mono
tff(fact_2129_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_2130_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),N))) ).

% Suc_div_le_mono
tff(fact_2131_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( ! [A: bool,B: bool] :
            ( ( X = vEBT_Leaf(A,B) )
           => ~ ( ( pp(A)
                 => ( Y = some_nat(zero_zero_nat) ) )
                & ( ~ pp(A)
                 => ( ( pp(B)
                     => ( Y = some_nat(one_one_nat) ) )
                    & ( ~ pp(B)
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != some_nat(Mi2) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_2132_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( ! [A: bool,B: bool] :
            ( ( X = vEBT_Leaf(A,B) )
           => ~ ( ( pp(B)
                 => ( Y = some_nat(one_one_nat) ) )
                & ( ~ pp(B)
                 => ( ( pp(A)
                     => ( Y = some_nat(zero_zero_nat) ) )
                    & ( ~ pp(A)
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != some_nat(Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_2133_diff__shunt__var,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( minus_minus_set_real(X,Y) = bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(X),Y)) ) ).

% diff_shunt_var
tff(fact_2134_diff__shunt__var,axiom,
    ! [X: set_bool,Y: set_bool] :
      ( ( minus_minus_set_bool(X,Y) = bot_bot_set_bool )
    <=> pp(aa_set_bool_bool(ord_less_eq_set_bool(X),Y)) ) ).

% diff_shunt_var
tff(fact_2135_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X),Y) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y)) ) ).

% diff_shunt_var
tff(fact_2136_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,X),Y) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).

% diff_shunt_var
tff(fact_2137_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% div_greater_zero_iff
tff(fact_2138_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M))) ) ) ).

% div_le_mono2
tff(fact_2139_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = M )
      <=> ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_2140_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M)) ) ) ).

% div_less_dividend
tff(fact_2141_pos__imp__zdiv__neg__iff,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)),zero_zero_int))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_2142_neg__imp__zdiv__neg__iff,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)),zero_zero_int))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2)) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_2143_div__neg__pos__less0,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)),zero_zero_int)) ) ) ).

% div_neg_pos_less0
tff(fact_2144_div__positive,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B3))) ) ) ).

% div_positive
tff(fact_2145_div__positive,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3))) ) ) ).

% div_positive
tff(fact_2146_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B3) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2147_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2148_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

% div_geq
tff(fact_2149_div__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

% div_if
tff(fact_2150_zdiv__mono1,axiom,
    ! [A2: int,A5: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),A5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A5),B3))) ) ) ).

% zdiv_mono1
tff(fact_2151_zdiv__mono2,axiom,
    ! [A2: int,B5: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),B3))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B5))) ) ) ) ).

% zdiv_mono2
tff(fact_2152_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_2153_zdiv__mono1__neg,axiom,
    ! [A2: int,A5: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),A5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A5),B3)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3))) ) ) ).

% zdiv_mono1_neg
tff(fact_2154_zdiv__mono2__neg,axiom,
    ! [A2: int,B5: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),B3))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B5)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_2155_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ) ).

% div_int_pos_iff
tff(fact_2156_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L))) ) ) ).

% div_positive_int
tff(fact_2157_div__nonneg__neg__le0,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)),zero_zero_int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_2158_div__nonpos__pos__le0,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)),zero_zero_int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_2159_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_2160_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_2161_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2)) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_2162_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)))
      <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3)) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_2163_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),K)),X)) ) ) ).

% int_div_less_self
tff(fact_2164_delete__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T,X)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,vEBT_set_vebt(T)),insert_nat(X,bot_bot_set_nat)) ) ) ).

% delete_correct
tff(fact_2165_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),Deg))
     => ( ( Mi = Ma )
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
             => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_13)) )
          & ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_13)) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_2166_divides__aux__eq,axiom,
    ! [Q3: nat,R2: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R2)))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_2167_divides__aux__eq,axiom,
    ! [Q3: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(unique6319869463603278526ux_int,product_Pair_int_int(Q3,R2)))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_2168_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T,X)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,vEBT_VEBT_set_vebt(T)),insert_nat(X,bot_bot_set_nat)) ) ) ).

% delete_correct'
tff(fact_2169_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_2170_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_2171_dbl__dec__simps_I2_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,zero_zero_rat) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_2172_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_2173_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_2174_prod__decode__aux_Osimps,axiom,
    ! [M: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),M)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,K))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_2175_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(X),Xa2) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
         => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
         => ( Y = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_2176_insert__absorb2,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(X,insert8211810215607154385at_nat(X,A3)) = insert8211810215607154385at_nat(X,A3) ).

% insert_absorb2
tff(fact_2177_insert__absorb2,axiom,
    ! [X: real,A3: set_real] : insert_real(X,insert_real(X,A3)) = insert_real(X,A3) ).

% insert_absorb2
tff(fact_2178_insert__absorb2,axiom,
    ! [X: bool,A3: set_bool] : insert_bool(X,insert_bool(X,A3)) = insert_bool(X,A3) ).

% insert_absorb2
tff(fact_2179_insert__absorb2,axiom,
    ! [X: nat,A3: set_nat] : insert_nat(X,insert_nat(X,A3)) = insert_nat(X,A3) ).

% insert_absorb2
tff(fact_2180_insert__absorb2,axiom,
    ! [X: int,A3: set_int] : insert_int(X,insert_int(X,A3)) = insert_int(X,A3) ).

% insert_absorb2
tff(fact_2181_insert__iff,axiom,
    ! [A2: product_prod_nat_nat,B3: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),insert8211810215607154385at_nat(B3,A3)))
    <=> ( ( A2 = B3 )
        | pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A3)) ) ) ).

% insert_iff
tff(fact_2182_insert__iff,axiom,
    ! [A2: real,B3: real,A3: set_real] :
      ( pp(aa_set_real_bool(member_real(A2),insert_real(B3,A3)))
    <=> ( ( A2 = B3 )
        | pp(aa_set_real_bool(member_real(A2),A3)) ) ) ).

% insert_iff
tff(fact_2183_insert__iff,axiom,
    ! [A2: bool,B3: bool,A3: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(A2),insert_bool(B3,A3)))
    <=> ( ( pp(A2)
        <=> pp(B3) )
        | pp(aa_set_bool_bool(member_bool(A2),A3)) ) ) ).

% insert_iff
tff(fact_2184_insert__iff,axiom,
    ! [A2: set_nat,B3: set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A2),insert_set_nat(B3,A3)))
    <=> ( ( A2 = B3 )
        | pp(aa_set_set_nat_bool(member_set_nat(A2),A3)) ) ) ).

% insert_iff
tff(fact_2185_insert__iff,axiom,
    ! [A2: nat,B3: nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A2),insert_nat(B3,A3)))
    <=> ( ( A2 = B3 )
        | pp(aa_set_nat_bool(member_nat(A2),A3)) ) ) ).

% insert_iff
tff(fact_2186_insert__iff,axiom,
    ! [A2: int,B3: int,A3: set_int] :
      ( pp(aa_set_int_bool(member_int(A2),insert_int(B3,A3)))
    <=> ( ( A2 = B3 )
        | pp(aa_set_int_bool(member_int(A2),A3)) ) ) ).

% insert_iff
tff(fact_2187_insertCI,axiom,
    ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
      ( ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),B2))
       => ( A2 = B3 ) )
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),insert8211810215607154385at_nat(B3,B2))) ) ).

% insertCI
tff(fact_2188_insertCI,axiom,
    ! [A2: real,B2: set_real,B3: real] :
      ( ( ~ pp(aa_set_real_bool(member_real(A2),B2))
       => ( A2 = B3 ) )
     => pp(aa_set_real_bool(member_real(A2),insert_real(B3,B2))) ) ).

% insertCI
tff(fact_2189_insertCI,axiom,
    ! [A2: bool,B2: set_bool,B3: bool] :
      ( ( ~ pp(aa_set_bool_bool(member_bool(A2),B2))
       => ( pp(A2)
        <=> pp(B3) ) )
     => pp(aa_set_bool_bool(member_bool(A2),insert_bool(B3,B2))) ) ).

% insertCI
tff(fact_2190_insertCI,axiom,
    ! [A2: set_nat,B2: set_set_nat,B3: set_nat] :
      ( ( ~ pp(aa_set_set_nat_bool(member_set_nat(A2),B2))
       => ( A2 = B3 ) )
     => pp(aa_set_set_nat_bool(member_set_nat(A2),insert_set_nat(B3,B2))) ) ).

% insertCI
tff(fact_2191_insertCI,axiom,
    ! [A2: nat,B2: set_nat,B3: nat] :
      ( ( ~ pp(aa_set_nat_bool(member_nat(A2),B2))
       => ( A2 = B3 ) )
     => pp(aa_set_nat_bool(member_nat(A2),insert_nat(B3,B2))) ) ).

% insertCI
tff(fact_2192_insertCI,axiom,
    ! [A2: int,B2: set_int,B3: int] :
      ( ( ~ pp(aa_set_int_bool(member_int(A2),B2))
       => ( A2 = B3 ) )
     => pp(aa_set_int_bool(member_int(A2),insert_int(B3,B2))) ) ).

% insertCI
tff(fact_2193_singletonI,axiom,
    ! [A2: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat))) ).

% singletonI
tff(fact_2194_singletonI,axiom,
    ! [A2: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(A2),insert_set_nat(A2,bot_bot_set_set_nat))) ).

% singletonI
tff(fact_2195_singletonI,axiom,
    ! [A2: real] : pp(aa_set_real_bool(member_real(A2),insert_real(A2,bot_bot_set_real))) ).

% singletonI
tff(fact_2196_singletonI,axiom,
    ! [A2: bool] : pp(aa_set_bool_bool(member_bool(A2),insert_bool(A2,bot_bot_set_bool))) ).

% singletonI
tff(fact_2197_singletonI,axiom,
    ! [A2: nat] : pp(aa_set_nat_bool(member_nat(A2),insert_nat(A2,bot_bot_set_nat))) ).

% singletonI
tff(fact_2198_singletonI,axiom,
    ! [A2: int] : pp(aa_set_int_bool(member_int(A2),insert_int(A2,bot_bot_set_int))) ).

% singletonI
tff(fact_2199_finite__insert,axiom,
    ! [A2: real,A3: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,insert_real(A2,A3)))
    <=> pp(aa_set_real_bool(finite_finite_real,A3)) ) ).

% finite_insert
tff(fact_2200_finite__insert,axiom,
    ! [A2: bool,A3: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,insert_bool(A2,A3)))
    <=> pp(aa_set_bool_bool(finite_finite_bool,A3)) ) ).

% finite_insert
tff(fact_2201_finite__insert,axiom,
    ! [A2: nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,insert_nat(A2,A3)))
    <=> pp(aa_set_nat_bool(finite_finite_nat,A3)) ) ).

% finite_insert
tff(fact_2202_finite__insert,axiom,
    ! [A2: int,A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,insert_int(A2,A3)))
    <=> pp(aa_set_int_bool(finite_finite_int,A3)) ) ).

% finite_insert
tff(fact_2203_finite__insert,axiom,
    ! [A2: complex,A3: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,insert_complex(A2,A3)))
    <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,A3)) ) ).

% finite_insert
tff(fact_2204_finite__insert,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,insert8211810215607154385at_nat(A2,A3)))
    <=> pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3)) ) ).

% finite_insert
tff(fact_2205_finite__insert,axiom,
    ! [A2: extended_enat,A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,insert_Extended_enat(A2,A3)))
    <=> pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3)) ) ).

% finite_insert
tff(fact_2206_insert__subset,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(insert8211810215607154385at_nat(X,A3)),B2))
    <=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
        & pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),B2)) ) ) ).

% insert_subset
tff(fact_2207_insert__subset,axiom,
    ! [X: real,A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(insert_real(X,A3)),B2))
    <=> ( pp(aa_set_real_bool(member_real(X),B2))
        & pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2)) ) ) ).

% insert_subset
tff(fact_2208_insert__subset,axiom,
    ! [X: bool,A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(insert_bool(X,A3)),B2))
    <=> ( pp(aa_set_bool_bool(member_bool(X),B2))
        & pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),B2)) ) ) ).

% insert_subset
tff(fact_2209_insert__subset,axiom,
    ! [X: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(insert_set_nat(X,A3)),B2))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(X),B2))
        & pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),B2)) ) ) ).

% insert_subset
tff(fact_2210_insert__subset,axiom,
    ! [X: nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,insert_nat(X,A3)),B2))
    <=> ( pp(aa_set_nat_bool(member_nat(X),B2))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2)) ) ) ).

% insert_subset
tff(fact_2211_insert__subset,axiom,
    ! [X: int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(insert_int(X,A3)),B2))
    <=> ( pp(aa_set_int_bool(member_int(X),B2))
        & pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2)) ) ) ).

% insert_subset
tff(fact_2212_insert__Diff1,axiom,
    ! [X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X,A3)),B2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2) ) ) ).

% insert_Diff1
tff(fact_2213_insert__Diff1,axiom,
    ! [X: real,B2: set_real,A3: set_real] :
      ( pp(aa_set_real_bool(member_real(X),B2))
     => ( minus_minus_set_real(insert_real(X,A3),B2) = minus_minus_set_real(A3,B2) ) ) ).

% insert_Diff1
tff(fact_2214_insert__Diff1,axiom,
    ! [X: bool,B2: set_bool,A3: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(X),B2))
     => ( minus_minus_set_bool(insert_bool(X,A3),B2) = minus_minus_set_bool(A3,B2) ) ) ).

% insert_Diff1
tff(fact_2215_insert__Diff1,axiom,
    ! [X: set_nat,B2: set_set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),B2))
     => ( minus_2163939370556025621et_nat(insert_set_nat(X,A3),B2) = minus_2163939370556025621et_nat(A3,B2) ) ) ).

% insert_Diff1
tff(fact_2216_insert__Diff1,axiom,
    ! [X: int,B2: set_int,A3: set_int] :
      ( pp(aa_set_int_bool(member_int(X),B2))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,insert_int(X,A3)),B2) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2) ) ) ).

% insert_Diff1
tff(fact_2217_insert__Diff1,axiom,
    ! [X: nat,B2: set_nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),B2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(X,A3)),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2) ) ) ).

% insert_Diff1
tff(fact_2218_Diff__insert0,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(X,B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2) ) ) ).

% Diff_insert0
tff(fact_2219_Diff__insert0,axiom,
    ! [X: real,A3: set_real,B2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X),A3))
     => ( minus_minus_set_real(A3,insert_real(X,B2)) = minus_minus_set_real(A3,B2) ) ) ).

% Diff_insert0
tff(fact_2220_Diff__insert0,axiom,
    ! [X: bool,A3: set_bool,B2: set_bool] :
      ( ~ pp(aa_set_bool_bool(member_bool(X),A3))
     => ( minus_minus_set_bool(A3,insert_bool(X,B2)) = minus_minus_set_bool(A3,B2) ) ) ).

% Diff_insert0
tff(fact_2221_Diff__insert0,axiom,
    ! [X: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A3))
     => ( minus_2163939370556025621et_nat(A3,insert_set_nat(X,B2)) = minus_2163939370556025621et_nat(A3,B2) ) ) ).

% Diff_insert0
tff(fact_2222_Diff__insert0,axiom,
    ! [X: int,A3: set_int,B2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X),A3))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(X,B2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2) ) ) ).

% Diff_insert0
tff(fact_2223_Diff__insert0,axiom,
    ! [X: nat,A3: set_nat,B2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),A3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(X,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2) ) ) ).

% Diff_insert0
tff(fact_2224_singleton__insert__inj__eq,axiom,
    ! [B3: product_prod_nat_nat,A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat(B3,bot_bo2099793752762293965at_nat) = insert8211810215607154385at_nat(A2,A3) )
    <=> ( ( A2 = B3 )
        & pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),insert8211810215607154385at_nat(B3,bot_bo2099793752762293965at_nat))) ) ) ).

% singleton_insert_inj_eq
tff(fact_2225_singleton__insert__inj__eq,axiom,
    ! [B3: real,A2: real,A3: set_real] :
      ( ( insert_real(B3,bot_bot_set_real) = insert_real(A2,A3) )
    <=> ( ( A2 = B3 )
        & pp(aa_set_real_bool(ord_less_eq_set_real(A3),insert_real(B3,bot_bot_set_real))) ) ) ).

% singleton_insert_inj_eq
tff(fact_2226_singleton__insert__inj__eq,axiom,
    ! [B3: bool,A2: bool,A3: set_bool] :
      ( ( insert_bool(B3,bot_bot_set_bool) = insert_bool(A2,A3) )
    <=> ( ( pp(A2)
        <=> pp(B3) )
        & pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),insert_bool(B3,bot_bot_set_bool))) ) ) ).

% singleton_insert_inj_eq
tff(fact_2227_singleton__insert__inj__eq,axiom,
    ! [B3: nat,A2: nat,A3: set_nat] :
      ( ( insert_nat(B3,bot_bot_set_nat) = insert_nat(A2,A3) )
    <=> ( ( A2 = B3 )
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),insert_nat(B3,bot_bot_set_nat))) ) ) ).

% singleton_insert_inj_eq
tff(fact_2228_singleton__insert__inj__eq,axiom,
    ! [B3: int,A2: int,A3: set_int] :
      ( ( insert_int(B3,bot_bot_set_int) = insert_int(A2,A3) )
    <=> ( ( A2 = B3 )
        & pp(aa_set_int_bool(ord_less_eq_set_int(A3),insert_int(B3,bot_bot_set_int))) ) ) ).

% singleton_insert_inj_eq
tff(fact_2229_singleton__insert__inj__eq_H,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A2,A3) = insert8211810215607154385at_nat(B3,bot_bo2099793752762293965at_nat) )
    <=> ( ( A2 = B3 )
        & pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),insert8211810215607154385at_nat(B3,bot_bo2099793752762293965at_nat))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2230_singleton__insert__inj__eq_H,axiom,
    ! [A2: real,A3: set_real,B3: real] :
      ( ( insert_real(A2,A3) = insert_real(B3,bot_bot_set_real) )
    <=> ( ( A2 = B3 )
        & pp(aa_set_real_bool(ord_less_eq_set_real(A3),insert_real(B3,bot_bot_set_real))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2231_singleton__insert__inj__eq_H,axiom,
    ! [A2: bool,A3: set_bool,B3: bool] :
      ( ( insert_bool(A2,A3) = insert_bool(B3,bot_bot_set_bool) )
    <=> ( ( pp(A2)
        <=> pp(B3) )
        & pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),insert_bool(B3,bot_bot_set_bool))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2232_singleton__insert__inj__eq_H,axiom,
    ! [A2: nat,A3: set_nat,B3: nat] :
      ( ( insert_nat(A2,A3) = insert_nat(B3,bot_bot_set_nat) )
    <=> ( ( A2 = B3 )
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),insert_nat(B3,bot_bot_set_nat))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2233_singleton__insert__inj__eq_H,axiom,
    ! [A2: int,A3: set_int,B3: int] :
      ( ( insert_int(A2,A3) = insert_int(B3,bot_bot_set_int) )
    <=> ( ( A2 = B3 )
        & pp(aa_set_int_bool(ord_less_eq_set_int(A3),insert_int(B3,bot_bot_set_int))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2234_insert__Diff__single,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat))) = insert8211810215607154385at_nat(A2,A3) ).

% insert_Diff_single
tff(fact_2235_insert__Diff__single,axiom,
    ! [A2: real,A3: set_real] : insert_real(A2,minus_minus_set_real(A3,insert_real(A2,bot_bot_set_real))) = insert_real(A2,A3) ).

% insert_Diff_single
tff(fact_2236_insert__Diff__single,axiom,
    ! [A2: bool,A3: set_bool] : insert_bool(A2,minus_minus_set_bool(A3,insert_bool(A2,bot_bot_set_bool))) = insert_bool(A2,A3) ).

% insert_Diff_single
tff(fact_2237_insert__Diff__single,axiom,
    ! [A2: int,A3: set_int] : insert_int(A2,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(A2,bot_bot_set_int))) = insert_int(A2,A3) ).

% insert_Diff_single
tff(fact_2238_insert__Diff__single,axiom,
    ! [A2: nat,A3: set_nat] : insert_nat(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(A2,bot_bot_set_nat))) = insert_nat(A2,A3) ).

% insert_Diff_single
tff(fact_2239_finite__Diff__insert,axiom,
    ! [A3: set_real,A2: real,B2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A3,insert_real(A2,B2))))
    <=> pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A3,B2))) ) ).

% finite_Diff_insert
tff(fact_2240_finite__Diff__insert,axiom,
    ! [A3: set_bool,A2: bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,minus_minus_set_bool(A3,insert_bool(A2,B2))))
    <=> pp(aa_set_bool_bool(finite_finite_bool,minus_minus_set_bool(A3,B2))) ) ).

% finite_Diff_insert
tff(fact_2241_finite__Diff__insert,axiom,
    ! [A3: set_int,A2: int,B2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(A2,B2))))
    <=> pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2))) ) ).

% finite_Diff_insert
tff(fact_2242_finite__Diff__insert,axiom,
    ! [A3: set_complex,A2: complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A3,insert_complex(A2,B2))))
    <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A3,B2))) ) ).

% finite_Diff_insert
tff(fact_2243_finite__Diff__insert,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(A2,B2))))
    <=> pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2))) ) ).

% finite_Diff_insert
tff(fact_2244_finite__Diff__insert,axiom,
    ! [A3: set_Extended_enat,A2: extended_enat,B2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,minus_925952699566721837d_enat(A3,insert_Extended_enat(A2,B2))))
    <=> pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,minus_925952699566721837d_enat(A3,B2))) ) ).

% finite_Diff_insert
tff(fact_2245_finite__Diff__insert,axiom,
    ! [A3: set_nat,A2: nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(A2,B2))))
    <=> pp(aa_set_nat_bool(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2))) ) ).

% finite_Diff_insert
tff(fact_2246_card__insert__disjoint,axiom,
    ! [A3: set_real,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ~ pp(aa_set_real_bool(member_real(X),A3))
       => ( finite_card_real(insert_real(X,A3)) = aa_nat_nat(suc,finite_card_real(A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_2247_card__insert__disjoint,axiom,
    ! [A3: set_bool,X: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( ~ pp(aa_set_bool_bool(member_bool(X),A3))
       => ( finite_card_bool(insert_bool(X,A3)) = aa_nat_nat(suc,finite_card_bool(A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_2248_card__insert__disjoint,axiom,
    ! [A3: set_list_nat,X: list_nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( ~ member_list_nat(X,A3)
       => ( finite_card_list_nat(insert_list_nat(X,A3)) = aa_nat_nat(suc,finite_card_list_nat(A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_2249_card__insert__disjoint,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A3))
       => ( finite_card_set_nat(insert_set_nat(X,A3)) = aa_nat_nat(suc,finite_card_set_nat(A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_2250_card__insert__disjoint,axiom,
    ! [A3: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),A3))
       => ( aa_set_nat_nat(finite_card_nat,insert_nat(X,A3)) = aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_2251_card__insert__disjoint,axiom,
    ! [A3: set_int,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ~ pp(aa_set_int_bool(member_int(X),A3))
       => ( finite_card_int(insert_int(X,A3)) = aa_nat_nat(suc,finite_card_int(A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_2252_card__insert__disjoint,axiom,
    ! [A3: set_complex,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A3))
       => ( finite_card_complex(insert_complex(X,A3)) = aa_nat_nat(suc,finite_card_complex(A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_2253_card__insert__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
       => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,insert8211810215607154385at_nat(X,A3)) = aa_nat_nat(suc,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_2254_card__insert__disjoint,axiom,
    ! [A3: set_Extended_enat,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A3))
       => ( finite121521170596916366d_enat(insert_Extended_enat(X,A3)) = aa_nat_nat(suc,finite121521170596916366d_enat(A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_2255_subset__Compl__singleton,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),uminus6524753893492686040at_nat(insert8211810215607154385at_nat(B3,bot_bo2099793752762293965at_nat))))
    <=> ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B3),A3)) ) ).

% subset_Compl_singleton
tff(fact_2256_subset__Compl__singleton,axiom,
    ! [A3: set_set_nat,B3: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),uminus613421341184616069et_nat(insert_set_nat(B3,bot_bot_set_set_nat))))
    <=> ~ pp(aa_set_set_nat_bool(member_set_nat(B3),A3)) ) ).

% subset_Compl_singleton
tff(fact_2257_subset__Compl__singleton,axiom,
    ! [A3: set_real,B3: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),uminus612125837232591019t_real(insert_real(B3,bot_bot_set_real))))
    <=> ~ pp(aa_set_real_bool(member_real(B3),A3)) ) ).

% subset_Compl_singleton
tff(fact_2258_subset__Compl__singleton,axiom,
    ! [A3: set_bool,B3: bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),uminus5166143097222041326t_bool(insert_bool(B3,bot_bot_set_bool))))
    <=> ~ pp(aa_set_bool_bool(member_bool(B3),A3)) ) ).

% subset_Compl_singleton
tff(fact_2259_subset__Compl__singleton,axiom,
    ! [A3: set_nat,B3: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),uminus5710092332889474511et_nat(insert_nat(B3,bot_bot_set_nat))))
    <=> ~ pp(aa_set_nat_bool(member_nat(B3),A3)) ) ).

% subset_Compl_singleton
tff(fact_2260_subset__Compl__singleton,axiom,
    ! [A3: set_int,B3: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),uminus1532241313380277803et_int(insert_int(B3,bot_bot_set_int))))
    <=> ~ pp(aa_set_int_bool(member_int(B3),A3)) ) ).

% subset_Compl_singleton
tff(fact_2261_card__Diff__insert,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A3))
     => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),B2))
       => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(A2,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_2262_card__Diff__insert,axiom,
    ! [A2: real,A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(A2),A3))
     => ( ~ pp(aa_set_real_bool(member_real(A2),B2))
       => ( finite_card_real(minus_minus_set_real(A3,insert_real(A2,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_real(minus_minus_set_real(A3,B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_2263_card__Diff__insert,axiom,
    ! [A2: bool,A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(A2),A3))
     => ( ~ pp(aa_set_bool_bool(member_bool(A2),B2))
       => ( finite_card_bool(minus_minus_set_bool(A3,insert_bool(A2,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_bool(minus_minus_set_bool(A3,B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_2264_card__Diff__insert,axiom,
    ! [A2: complex,A3: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(member_complex(A2),A3))
     => ( ~ pp(aa_set_complex_bool(member_complex(A2),B2))
       => ( finite_card_complex(minus_811609699411566653omplex(A3,insert_complex(A2,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(minus_811609699411566653omplex(A3,B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_2265_card__Diff__insert,axiom,
    ! [A2: list_nat,A3: set_list_nat,B2: set_list_nat] :
      ( member_list_nat(A2,A3)
     => ( ~ member_list_nat(A2,B2)
       => ( finite_card_list_nat(minus_7954133019191499631st_nat(A3,insert_list_nat(A2,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A3,B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_2266_card__Diff__insert,axiom,
    ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A2),A3))
     => ( ~ pp(aa_set_set_nat_bool(member_set_nat(A2),B2))
       => ( finite_card_set_nat(minus_2163939370556025621et_nat(A3,insert_set_nat(A2,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A3,B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_2267_card__Diff__insert,axiom,
    ! [A2: int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(A2),A3))
     => ( ~ pp(aa_set_int_bool(member_int(A2),B2))
       => ( finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(A2,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_2268_card__Diff__insert,axiom,
    ! [A2: nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A2),A3))
     => ( ~ pp(aa_set_nat_bool(member_nat(A2),B2))
       => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(A2,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_2269_less__eq__real__def,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% less_eq_real_def
tff(fact_2270_complete__real,axiom,
    ! [S: set_real] :
      ( ? [X4: real] : pp(aa_set_real_bool(member_real(X4),S))
     => ( ? [Z4: real] :
          ! [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),Z4)) )
       => ? [Y3: real] :
            ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3)) )
            & ! [Z4: real] :
                ( ! [X5: real] :
                    ( pp(aa_set_real_bool(member_real(X5),S))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),Z4)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),Z4)) ) ) ) ) ).

% complete_real
tff(fact_2271_mk__disjoint__insert,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A3))
     => ? [B7: set_Pr1261947904930325089at_nat] :
          ( ( A3 = insert8211810215607154385at_nat(A2,B7) )
          & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),B7)) ) ) ).

% mk_disjoint_insert
tff(fact_2272_mk__disjoint__insert,axiom,
    ! [A2: real,A3: set_real] :
      ( pp(aa_set_real_bool(member_real(A2),A3))
     => ? [B7: set_real] :
          ( ( A3 = insert_real(A2,B7) )
          & ~ pp(aa_set_real_bool(member_real(A2),B7)) ) ) ).

% mk_disjoint_insert
tff(fact_2273_mk__disjoint__insert,axiom,
    ! [A2: bool,A3: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(A2),A3))
     => ? [B7: set_bool] :
          ( ( A3 = insert_bool(A2,B7) )
          & ~ pp(aa_set_bool_bool(member_bool(A2),B7)) ) ) ).

% mk_disjoint_insert
tff(fact_2274_mk__disjoint__insert,axiom,
    ! [A2: set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A2),A3))
     => ? [B7: set_set_nat] :
          ( ( A3 = insert_set_nat(A2,B7) )
          & ~ pp(aa_set_set_nat_bool(member_set_nat(A2),B7)) ) ) ).

% mk_disjoint_insert
tff(fact_2275_mk__disjoint__insert,axiom,
    ! [A2: nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A2),A3))
     => ? [B7: set_nat] :
          ( ( A3 = insert_nat(A2,B7) )
          & ~ pp(aa_set_nat_bool(member_nat(A2),B7)) ) ) ).

% mk_disjoint_insert
tff(fact_2276_mk__disjoint__insert,axiom,
    ! [A2: int,A3: set_int] :
      ( pp(aa_set_int_bool(member_int(A2),A3))
     => ? [B7: set_int] :
          ( ( A3 = insert_int(A2,B7) )
          & ~ pp(aa_set_int_bool(member_int(A2),B7)) ) ) ).

% mk_disjoint_insert
tff(fact_2277_insert__commute,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(X,insert8211810215607154385at_nat(Y,A3)) = insert8211810215607154385at_nat(Y,insert8211810215607154385at_nat(X,A3)) ).

% insert_commute
tff(fact_2278_insert__commute,axiom,
    ! [X: real,Y: real,A3: set_real] : insert_real(X,insert_real(Y,A3)) = insert_real(Y,insert_real(X,A3)) ).

% insert_commute
tff(fact_2279_insert__commute,axiom,
    ! [X: bool,Y: bool,A3: set_bool] : insert_bool(X,insert_bool(Y,A3)) = insert_bool(Y,insert_bool(X,A3)) ).

% insert_commute
tff(fact_2280_insert__commute,axiom,
    ! [X: nat,Y: nat,A3: set_nat] : insert_nat(X,insert_nat(Y,A3)) = insert_nat(Y,insert_nat(X,A3)) ).

% insert_commute
tff(fact_2281_insert__commute,axiom,
    ! [X: int,Y: int,A3: set_int] : insert_int(X,insert_int(Y,A3)) = insert_int(Y,insert_int(X,A3)) ).

% insert_commute
tff(fact_2282_insert__eq__iff,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A3))
     => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B3),B2))
       => ( ( insert8211810215607154385at_nat(A2,A3) = insert8211810215607154385at_nat(B3,B2) )
        <=> ( ( ( A2 = B3 )
             => ( A3 = B2 ) )
            & ( ( A2 != B3 )
             => ? [C4: set_Pr1261947904930325089at_nat] :
                  ( ( A3 = insert8211810215607154385at_nat(B3,C4) )
                  & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B3),C4))
                  & ( B2 = insert8211810215607154385at_nat(A2,C4) )
                  & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),C4)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_2283_insert__eq__iff,axiom,
    ! [A2: real,A3: set_real,B3: real,B2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(A2),A3))
     => ( ~ pp(aa_set_real_bool(member_real(B3),B2))
       => ( ( insert_real(A2,A3) = insert_real(B3,B2) )
        <=> ( ( ( A2 = B3 )
             => ( A3 = B2 ) )
            & ( ( A2 != B3 )
             => ? [C4: set_real] :
                  ( ( A3 = insert_real(B3,C4) )
                  & ~ pp(aa_set_real_bool(member_real(B3),C4))
                  & ( B2 = insert_real(A2,C4) )
                  & ~ pp(aa_set_real_bool(member_real(A2),C4)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_2284_insert__eq__iff,axiom,
    ! [A2: bool,A3: set_bool,B3: bool,B2: set_bool] :
      ( ~ pp(aa_set_bool_bool(member_bool(A2),A3))
     => ( ~ pp(aa_set_bool_bool(member_bool(B3),B2))
       => ( ( insert_bool(A2,A3) = insert_bool(B3,B2) )
        <=> ( ( ( pp(A2)
              <=> pp(B3) )
             => ( A3 = B2 ) )
            & ( ( pp(A2)
              <=> ~ pp(B3) )
             => ? [C4: set_bool] :
                  ( ( A3 = insert_bool(B3,C4) )
                  & ~ pp(aa_set_bool_bool(member_bool(B3),C4))
                  & ( B2 = insert_bool(A2,C4) )
                  & ~ pp(aa_set_bool_bool(member_bool(A2),C4)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_2285_insert__eq__iff,axiom,
    ! [A2: set_nat,A3: set_set_nat,B3: set_nat,B2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(A2),A3))
     => ( ~ pp(aa_set_set_nat_bool(member_set_nat(B3),B2))
       => ( ( insert_set_nat(A2,A3) = insert_set_nat(B3,B2) )
        <=> ( ( ( A2 = B3 )
             => ( A3 = B2 ) )
            & ( ( A2 != B3 )
             => ? [C4: set_set_nat] :
                  ( ( A3 = insert_set_nat(B3,C4) )
                  & ~ pp(aa_set_set_nat_bool(member_set_nat(B3),C4))
                  & ( B2 = insert_set_nat(A2,C4) )
                  & ~ pp(aa_set_set_nat_bool(member_set_nat(A2),C4)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_2286_insert__eq__iff,axiom,
    ! [A2: nat,A3: set_nat,B3: nat,B2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(A2),A3))
     => ( ~ pp(aa_set_nat_bool(member_nat(B3),B2))
       => ( ( insert_nat(A2,A3) = insert_nat(B3,B2) )
        <=> ( ( ( A2 = B3 )
             => ( A3 = B2 ) )
            & ( ( A2 != B3 )
             => ? [C4: set_nat] :
                  ( ( A3 = insert_nat(B3,C4) )
                  & ~ pp(aa_set_nat_bool(member_nat(B3),C4))
                  & ( B2 = insert_nat(A2,C4) )
                  & ~ pp(aa_set_nat_bool(member_nat(A2),C4)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_2287_insert__eq__iff,axiom,
    ! [A2: int,A3: set_int,B3: int,B2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(A2),A3))
     => ( ~ pp(aa_set_int_bool(member_int(B3),B2))
       => ( ( insert_int(A2,A3) = insert_int(B3,B2) )
        <=> ( ( ( A2 = B3 )
             => ( A3 = B2 ) )
            & ( ( A2 != B3 )
             => ? [C4: set_int] :
                  ( ( A3 = insert_int(B3,C4) )
                  & ~ pp(aa_set_int_bool(member_int(B3),C4))
                  & ( B2 = insert_int(A2,C4) )
                  & ~ pp(aa_set_int_bool(member_int(A2),C4)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_2288_insert__absorb,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A3))
     => ( insert8211810215607154385at_nat(A2,A3) = A3 ) ) ).

% insert_absorb
tff(fact_2289_insert__absorb,axiom,
    ! [A2: real,A3: set_real] :
      ( pp(aa_set_real_bool(member_real(A2),A3))
     => ( insert_real(A2,A3) = A3 ) ) ).

% insert_absorb
tff(fact_2290_insert__absorb,axiom,
    ! [A2: bool,A3: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(A2),A3))
     => ( insert_bool(A2,A3) = A3 ) ) ).

% insert_absorb
tff(fact_2291_insert__absorb,axiom,
    ! [A2: set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A2),A3))
     => ( insert_set_nat(A2,A3) = A3 ) ) ).

% insert_absorb
tff(fact_2292_insert__absorb,axiom,
    ! [A2: nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A2),A3))
     => ( insert_nat(A2,A3) = A3 ) ) ).

% insert_absorb
tff(fact_2293_insert__absorb,axiom,
    ! [A2: int,A3: set_int] :
      ( pp(aa_set_int_bool(member_int(A2),A3))
     => ( insert_int(A2,A3) = A3 ) ) ).

% insert_absorb
tff(fact_2294_insert__ident,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
     => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
       => ( ( insert8211810215607154385at_nat(X,A3) = insert8211810215607154385at_nat(X,B2) )
        <=> ( A3 = B2 ) ) ) ) ).

% insert_ident
tff(fact_2295_insert__ident,axiom,
    ! [X: real,A3: set_real,B2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X),A3))
     => ( ~ pp(aa_set_real_bool(member_real(X),B2))
       => ( ( insert_real(X,A3) = insert_real(X,B2) )
        <=> ( A3 = B2 ) ) ) ) ).

% insert_ident
tff(fact_2296_insert__ident,axiom,
    ! [X: bool,A3: set_bool,B2: set_bool] :
      ( ~ pp(aa_set_bool_bool(member_bool(X),A3))
     => ( ~ pp(aa_set_bool_bool(member_bool(X),B2))
       => ( ( insert_bool(X,A3) = insert_bool(X,B2) )
        <=> ( A3 = B2 ) ) ) ) ).

% insert_ident
tff(fact_2297_insert__ident,axiom,
    ! [X: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A3))
     => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),B2))
       => ( ( insert_set_nat(X,A3) = insert_set_nat(X,B2) )
        <=> ( A3 = B2 ) ) ) ) ).

% insert_ident
tff(fact_2298_insert__ident,axiom,
    ! [X: nat,A3: set_nat,B2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),A3))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),B2))
       => ( ( insert_nat(X,A3) = insert_nat(X,B2) )
        <=> ( A3 = B2 ) ) ) ) ).

% insert_ident
tff(fact_2299_insert__ident,axiom,
    ! [X: int,A3: set_int,B2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X),A3))
     => ( ~ pp(aa_set_int_bool(member_int(X),B2))
       => ( ( insert_int(X,A3) = insert_int(X,B2) )
        <=> ( A3 = B2 ) ) ) ) ).

% insert_ident
tff(fact_2300_Set_Oset__insert,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
     => ~ ! [B7: set_Pr1261947904930325089at_nat] :
            ( ( A3 = insert8211810215607154385at_nat(X,B7) )
           => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B7)) ) ) ).

% Set.set_insert
tff(fact_2301_Set_Oset__insert,axiom,
    ! [X: real,A3: set_real] :
      ( pp(aa_set_real_bool(member_real(X),A3))
     => ~ ! [B7: set_real] :
            ( ( A3 = insert_real(X,B7) )
           => pp(aa_set_real_bool(member_real(X),B7)) ) ) ).

% Set.set_insert
tff(fact_2302_Set_Oset__insert,axiom,
    ! [X: bool,A3: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(X),A3))
     => ~ ! [B7: set_bool] :
            ( ( A3 = insert_bool(X,B7) )
           => pp(aa_set_bool_bool(member_bool(X),B7)) ) ) ).

% Set.set_insert
tff(fact_2303_Set_Oset__insert,axiom,
    ! [X: set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),A3))
     => ~ ! [B7: set_set_nat] :
            ( ( A3 = insert_set_nat(X,B7) )
           => pp(aa_set_set_nat_bool(member_set_nat(X),B7)) ) ) ).

% Set.set_insert
tff(fact_2304_Set_Oset__insert,axiom,
    ! [X: nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),A3))
     => ~ ! [B7: set_nat] :
            ( ( A3 = insert_nat(X,B7) )
           => pp(aa_set_nat_bool(member_nat(X),B7)) ) ) ).

% Set.set_insert
tff(fact_2305_Set_Oset__insert,axiom,
    ! [X: int,A3: set_int] :
      ( pp(aa_set_int_bool(member_int(X),A3))
     => ~ ! [B7: set_int] :
            ( ( A3 = insert_int(X,B7) )
           => pp(aa_set_int_bool(member_int(X),B7)) ) ) ).

% Set.set_insert
tff(fact_2306_insertI2,axiom,
    ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),B2))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),insert8211810215607154385at_nat(B3,B2))) ) ).

% insertI2
tff(fact_2307_insertI2,axiom,
    ! [A2: real,B2: set_real,B3: real] :
      ( pp(aa_set_real_bool(member_real(A2),B2))
     => pp(aa_set_real_bool(member_real(A2),insert_real(B3,B2))) ) ).

% insertI2
tff(fact_2308_insertI2,axiom,
    ! [A2: bool,B2: set_bool,B3: bool] :
      ( pp(aa_set_bool_bool(member_bool(A2),B2))
     => pp(aa_set_bool_bool(member_bool(A2),insert_bool(B3,B2))) ) ).

% insertI2
tff(fact_2309_insertI2,axiom,
    ! [A2: set_nat,B2: set_set_nat,B3: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A2),B2))
     => pp(aa_set_set_nat_bool(member_set_nat(A2),insert_set_nat(B3,B2))) ) ).

% insertI2
tff(fact_2310_insertI2,axiom,
    ! [A2: nat,B2: set_nat,B3: nat] :
      ( pp(aa_set_nat_bool(member_nat(A2),B2))
     => pp(aa_set_nat_bool(member_nat(A2),insert_nat(B3,B2))) ) ).

% insertI2
tff(fact_2311_insertI2,axiom,
    ! [A2: int,B2: set_int,B3: int] :
      ( pp(aa_set_int_bool(member_int(A2),B2))
     => pp(aa_set_int_bool(member_int(A2),insert_int(B3,B2))) ) ).

% insertI2
tff(fact_2312_insertI1,axiom,
    ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),insert8211810215607154385at_nat(A2,B2))) ).

% insertI1
tff(fact_2313_insertI1,axiom,
    ! [A2: real,B2: set_real] : pp(aa_set_real_bool(member_real(A2),insert_real(A2,B2))) ).

% insertI1
tff(fact_2314_insertI1,axiom,
    ! [A2: bool,B2: set_bool] : pp(aa_set_bool_bool(member_bool(A2),insert_bool(A2,B2))) ).

% insertI1
tff(fact_2315_insertI1,axiom,
    ! [A2: set_nat,B2: set_set_nat] : pp(aa_set_set_nat_bool(member_set_nat(A2),insert_set_nat(A2,B2))) ).

% insertI1
tff(fact_2316_insertI1,axiom,
    ! [A2: nat,B2: set_nat] : pp(aa_set_nat_bool(member_nat(A2),insert_nat(A2,B2))) ).

% insertI1
tff(fact_2317_insertI1,axiom,
    ! [A2: int,B2: set_int] : pp(aa_set_int_bool(member_int(A2),insert_int(A2,B2))) ).

% insertI1
tff(fact_2318_insertE,axiom,
    ! [A2: product_prod_nat_nat,B3: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),insert8211810215607154385at_nat(B3,A3)))
     => ( ( A2 != B3 )
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A3)) ) ) ).

% insertE
tff(fact_2319_insertE,axiom,
    ! [A2: real,B3: real,A3: set_real] :
      ( pp(aa_set_real_bool(member_real(A2),insert_real(B3,A3)))
     => ( ( A2 != B3 )
       => pp(aa_set_real_bool(member_real(A2),A3)) ) ) ).

% insertE
tff(fact_2320_insertE,axiom,
    ! [A2: bool,B3: bool,A3: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(A2),insert_bool(B3,A3)))
     => ( ( pp(A2)
        <=> ~ pp(B3) )
       => pp(aa_set_bool_bool(member_bool(A2),A3)) ) ) ).

% insertE
tff(fact_2321_insertE,axiom,
    ! [A2: set_nat,B3: set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A2),insert_set_nat(B3,A3)))
     => ( ( A2 != B3 )
       => pp(aa_set_set_nat_bool(member_set_nat(A2),A3)) ) ) ).

% insertE
tff(fact_2322_insertE,axiom,
    ! [A2: nat,B3: nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A2),insert_nat(B3,A3)))
     => ( ( A2 != B3 )
       => pp(aa_set_nat_bool(member_nat(A2),A3)) ) ) ).

% insertE
tff(fact_2323_insertE,axiom,
    ! [A2: int,B3: int,A3: set_int] :
      ( pp(aa_set_int_bool(member_int(A2),insert_int(B3,A3)))
     => ( ( A2 != B3 )
       => pp(aa_set_int_bool(member_int(A2),A3)) ) ) ).

% insertE
tff(fact_2324_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc7004362771917097900on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_nat,Uv2: option_nat] : X != produc6699827698519632094on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_nat,V2: nat] : X != produc6699827698519632094on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F3: fun_nat_fun_nat_nat,A: nat,B: nat] : X != produc6699827698519632094on_nat(F3,produc5098337634421038937on_nat(some_nat(A),some_nat(B))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_2325_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc1260151622200656695at_nat] :
      ( ! [Uu2: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : X != produc7814092408141220007at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr2477239776880934807at_nat,V2: product_prod_nat_nat] : X != produc7814092408141220007at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F3: fun_Pr2477239776880934807at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] : X != produc7814092408141220007at_nat(F3,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A),some_P7363390416028606310at_nat(B))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_2326_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc2703417854250991842on_num] :
      ( ! [Uu2: fun_num_fun_num_num,Uv2: option_num] : X != produc6360833703471310868on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_num,V2: num] : X != produc6360833703471310868on_num(Uw2,produc8585076106096196333on_num(some_num(V2),none_num))
       => ~ ! [F3: fun_num_fun_num_num,A: num,B: num] : X != produc6360833703471310868on_num(F3,produc8585076106096196333on_num(some_num(A),some_num(B))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_2327_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc3700678068921409119on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_bool,Uv2: option_nat] : X != produc2004789812759622991on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_bool,V2: nat] : X != produc2004789812759622991on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F3: fun_nat_fun_nat_bool,X5: nat,Y3: nat] : X != produc2004789812759622991on_nat(F3,produc5098337634421038937on_nat(some_nat(X5),some_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_2328_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc533602882674032673at_nat] :
      ( ! [Uu2: fun_Pr248683899324676269t_bool,Uv2: option4927543243414619207at_nat] : X != produc5961576194287291153at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr248683899324676269t_bool,V2: product_prod_nat_nat] : X != produc5961576194287291153at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F3: fun_Pr248683899324676269t_bool,X5: product_prod_nat_nat,Y3: product_prod_nat_nat] : X != produc5961576194287291153at_nat(F3,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X5),some_P7363390416028606310at_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_2329_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc6167219322466932359on_num] :
      ( ! [Uu2: fun_num_fun_num_bool,Uv2: option_num] : X != produc8433281988922931063on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_bool,V2: num] : X != produc8433281988922931063on_num(Uw2,produc8585076106096196333on_num(some_num(V2),none_num))
       => ~ ! [F3: fun_num_fun_num_bool,X5: num,Y3: num] : X != produc8433281988922931063on_num(F3,produc8585076106096196333on_num(some_num(X5),some_num(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_2330_singletonD,axiom,
    ! [B3: product_prod_nat_nat,A2: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B3),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat)))
     => ( B3 = A2 ) ) ).

% singletonD
tff(fact_2331_singletonD,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(B3),insert_set_nat(A2,bot_bot_set_set_nat)))
     => ( B3 = A2 ) ) ).

% singletonD
tff(fact_2332_singletonD,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_set_real_bool(member_real(B3),insert_real(A2,bot_bot_set_real)))
     => ( B3 = A2 ) ) ).

% singletonD
tff(fact_2333_singletonD,axiom,
    ! [B3: bool,A2: bool] :
      ( pp(aa_set_bool_bool(member_bool(B3),insert_bool(A2,bot_bot_set_bool)))
     => ( pp(B3)
      <=> pp(A2) ) ) ).

% singletonD
tff(fact_2334_singletonD,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_set_nat_bool(member_nat(B3),insert_nat(A2,bot_bot_set_nat)))
     => ( B3 = A2 ) ) ).

% singletonD
tff(fact_2335_singletonD,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_set_int_bool(member_int(B3),insert_int(A2,bot_bot_set_int)))
     => ( B3 = A2 ) ) ).

% singletonD
tff(fact_2336_singleton__iff,axiom,
    ! [B3: product_prod_nat_nat,A2: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B3),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat)))
    <=> ( B3 = A2 ) ) ).

% singleton_iff
tff(fact_2337_singleton__iff,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(B3),insert_set_nat(A2,bot_bot_set_set_nat)))
    <=> ( B3 = A2 ) ) ).

% singleton_iff
tff(fact_2338_singleton__iff,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_set_real_bool(member_real(B3),insert_real(A2,bot_bot_set_real)))
    <=> ( B3 = A2 ) ) ).

% singleton_iff
tff(fact_2339_singleton__iff,axiom,
    ! [B3: bool,A2: bool] :
      ( pp(aa_set_bool_bool(member_bool(B3),insert_bool(A2,bot_bot_set_bool)))
    <=> ( pp(B3)
      <=> pp(A2) ) ) ).

% singleton_iff
tff(fact_2340_singleton__iff,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_set_nat_bool(member_nat(B3),insert_nat(A2,bot_bot_set_nat)))
    <=> ( B3 = A2 ) ) ).

% singleton_iff
tff(fact_2341_singleton__iff,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_set_int_bool(member_int(B3),insert_int(A2,bot_bot_set_int)))
    <=> ( B3 = A2 ) ) ).

% singleton_iff
tff(fact_2342_doubleton__eq__iff,axiom,
    ! [A2: product_prod_nat_nat,B3: product_prod_nat_nat,C: product_prod_nat_nat,D: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A2,insert8211810215607154385at_nat(B3,bot_bo2099793752762293965at_nat)) = insert8211810215607154385at_nat(C,insert8211810215607154385at_nat(D,bot_bo2099793752762293965at_nat)) )
    <=> ( ( ( A2 = C )
          & ( B3 = D ) )
        | ( ( A2 = D )
          & ( B3 = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_2343_doubleton__eq__iff,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( insert_real(A2,insert_real(B3,bot_bot_set_real)) = insert_real(C,insert_real(D,bot_bot_set_real)) )
    <=> ( ( ( A2 = C )
          & ( B3 = D ) )
        | ( ( A2 = D )
          & ( B3 = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_2344_doubleton__eq__iff,axiom,
    ! [A2: bool,B3: bool,C: bool,D: bool] :
      ( ( insert_bool(A2,insert_bool(B3,bot_bot_set_bool)) = insert_bool(C,insert_bool(D,bot_bot_set_bool)) )
    <=> ( ( ( pp(A2)
          <=> pp(C) )
          & ( pp(B3)
          <=> pp(D) ) )
        | ( ( pp(A2)
          <=> pp(D) )
          & ( pp(B3)
          <=> pp(C) ) ) ) ) ).

% doubleton_eq_iff
tff(fact_2345_doubleton__eq__iff,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( ( insert_nat(A2,insert_nat(B3,bot_bot_set_nat)) = insert_nat(C,insert_nat(D,bot_bot_set_nat)) )
    <=> ( ( ( A2 = C )
          & ( B3 = D ) )
        | ( ( A2 = D )
          & ( B3 = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_2346_doubleton__eq__iff,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( insert_int(A2,insert_int(B3,bot_bot_set_int)) = insert_int(C,insert_int(D,bot_bot_set_int)) )
    <=> ( ( ( A2 = C )
          & ( B3 = D ) )
        | ( ( A2 = D )
          & ( B3 = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_2347_insert__not__empty,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A2,A3) != bot_bo2099793752762293965at_nat ).

% insert_not_empty
tff(fact_2348_insert__not__empty,axiom,
    ! [A2: real,A3: set_real] : insert_real(A2,A3) != bot_bot_set_real ).

% insert_not_empty
tff(fact_2349_insert__not__empty,axiom,
    ! [A2: bool,A3: set_bool] : insert_bool(A2,A3) != bot_bot_set_bool ).

% insert_not_empty
tff(fact_2350_insert__not__empty,axiom,
    ! [A2: nat,A3: set_nat] : insert_nat(A2,A3) != bot_bot_set_nat ).

% insert_not_empty
tff(fact_2351_insert__not__empty,axiom,
    ! [A2: int,A3: set_int] : insert_int(A2,A3) != bot_bot_set_int ).

% insert_not_empty
tff(fact_2352_singleton__inject,axiom,
    ! [A2: product_prod_nat_nat,B3: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat) = insert8211810215607154385at_nat(B3,bot_bo2099793752762293965at_nat) )
     => ( A2 = B3 ) ) ).

% singleton_inject
tff(fact_2353_singleton__inject,axiom,
    ! [A2: real,B3: real] :
      ( ( insert_real(A2,bot_bot_set_real) = insert_real(B3,bot_bot_set_real) )
     => ( A2 = B3 ) ) ).

% singleton_inject
tff(fact_2354_singleton__inject,axiom,
    ! [A2: bool,B3: bool] :
      ( ( insert_bool(A2,bot_bot_set_bool) = insert_bool(B3,bot_bot_set_bool) )
     => ( pp(A2)
      <=> pp(B3) ) ) ).

% singleton_inject
tff(fact_2355_singleton__inject,axiom,
    ! [A2: nat,B3: nat] :
      ( ( insert_nat(A2,bot_bot_set_nat) = insert_nat(B3,bot_bot_set_nat) )
     => ( A2 = B3 ) ) ).

% singleton_inject
tff(fact_2356_singleton__inject,axiom,
    ! [A2: int,B3: int] :
      ( ( insert_int(A2,bot_bot_set_int) = insert_int(B3,bot_bot_set_int) )
     => ( A2 = B3 ) ) ).

% singleton_inject
tff(fact_2357_finite_OinsertI,axiom,
    ! [A3: set_real,A2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => pp(aa_set_real_bool(finite_finite_real,insert_real(A2,A3))) ) ).

% finite.insertI
tff(fact_2358_finite_OinsertI,axiom,
    ! [A3: set_bool,A2: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => pp(aa_set_bool_bool(finite_finite_bool,insert_bool(A2,A3))) ) ).

% finite.insertI
tff(fact_2359_finite_OinsertI,axiom,
    ! [A3: set_nat,A2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => pp(aa_set_nat_bool(finite_finite_nat,insert_nat(A2,A3))) ) ).

% finite.insertI
tff(fact_2360_finite_OinsertI,axiom,
    ! [A3: set_int,A2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => pp(aa_set_int_bool(finite_finite_int,insert_int(A2,A3))) ) ).

% finite.insertI
tff(fact_2361_finite_OinsertI,axiom,
    ! [A3: set_complex,A2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,insert_complex(A2,A3))) ) ).

% finite.insertI
tff(fact_2362_finite_OinsertI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,insert8211810215607154385at_nat(A2,A3))) ) ).

% finite.insertI
tff(fact_2363_finite_OinsertI,axiom,
    ! [A3: set_Extended_enat,A2: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,insert_Extended_enat(A2,A3))) ) ).

% finite.insertI
tff(fact_2364_insert__mono,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,D4: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(C2),D4))
     => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(insert8211810215607154385at_nat(A2,C2)),insert8211810215607154385at_nat(A2,D4))) ) ).

% insert_mono
tff(fact_2365_insert__mono,axiom,
    ! [C2: set_real,D4: set_real,A2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(C2),D4))
     => pp(aa_set_real_bool(ord_less_eq_set_real(insert_real(A2,C2)),insert_real(A2,D4))) ) ).

% insert_mono
tff(fact_2366_insert__mono,axiom,
    ! [C2: set_bool,D4: set_bool,A2: bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(C2),D4))
     => pp(aa_set_bool_bool(ord_less_eq_set_bool(insert_bool(A2,C2)),insert_bool(A2,D4))) ) ).

% insert_mono
tff(fact_2367_insert__mono,axiom,
    ! [C2: set_nat,D4: set_nat,A2: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C2),D4))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,insert_nat(A2,C2)),insert_nat(A2,D4))) ) ).

% insert_mono
tff(fact_2368_insert__mono,axiom,
    ! [C2: set_int,D4: set_int,A2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),D4))
     => pp(aa_set_int_bool(ord_less_eq_set_int(insert_int(A2,C2)),insert_int(A2,D4))) ) ).

% insert_mono
tff(fact_2369_subset__insert,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),insert8211810215607154385at_nat(X,B2)))
      <=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),B2)) ) ) ).

% subset_insert
tff(fact_2370_subset__insert,axiom,
    ! [X: real,A3: set_real,B2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X),A3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),insert_real(X,B2)))
      <=> pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2)) ) ) ).

% subset_insert
tff(fact_2371_subset__insert,axiom,
    ! [X: bool,A3: set_bool,B2: set_bool] :
      ( ~ pp(aa_set_bool_bool(member_bool(X),A3))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),insert_bool(X,B2)))
      <=> pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),B2)) ) ) ).

% subset_insert
tff(fact_2372_subset__insert,axiom,
    ! [X: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A3))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),insert_set_nat(X,B2)))
      <=> pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),B2)) ) ) ).

% subset_insert
tff(fact_2373_subset__insert,axiom,
    ! [X: nat,A3: set_nat,B2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),A3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),insert_nat(X,B2)))
      <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2)) ) ) ).

% subset_insert
tff(fact_2374_subset__insert,axiom,
    ! [X: int,A3: set_int,B2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X),A3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),insert_int(X,B2)))
      <=> pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2)) ) ) ).

% subset_insert
tff(fact_2375_subset__insertI,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(B2),insert8211810215607154385at_nat(A2,B2))) ).

% subset_insertI
tff(fact_2376_subset__insertI,axiom,
    ! [B2: set_real,A2: real] : pp(aa_set_real_bool(ord_less_eq_set_real(B2),insert_real(A2,B2))) ).

% subset_insertI
tff(fact_2377_subset__insertI,axiom,
    ! [B2: set_bool,A2: bool] : pp(aa_set_bool_bool(ord_less_eq_set_bool(B2),insert_bool(A2,B2))) ).

% subset_insertI
tff(fact_2378_subset__insertI,axiom,
    ! [B2: set_nat,A2: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),insert_nat(A2,B2))) ).

% subset_insertI
tff(fact_2379_subset__insertI,axiom,
    ! [B2: set_int,A2: int] : pp(aa_set_int_bool(ord_less_eq_set_int(B2),insert_int(A2,B2))) ).

% subset_insertI
tff(fact_2380_subset__insertI2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),B2))
     => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),insert8211810215607154385at_nat(B3,B2))) ) ).

% subset_insertI2
tff(fact_2381_subset__insertI2,axiom,
    ! [A3: set_real,B2: set_real,B3: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2))
     => pp(aa_set_real_bool(ord_less_eq_set_real(A3),insert_real(B3,B2))) ) ).

% subset_insertI2
tff(fact_2382_subset__insertI2,axiom,
    ! [A3: set_bool,B2: set_bool,B3: bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),B2))
     => pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),insert_bool(B3,B2))) ) ).

% subset_insertI2
tff(fact_2383_subset__insertI2,axiom,
    ! [A3: set_nat,B2: set_nat,B3: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),insert_nat(B3,B2))) ) ).

% subset_insertI2
tff(fact_2384_subset__insertI2,axiom,
    ! [A3: set_int,B2: set_int,B3: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A3),insert_int(B3,B2))) ) ).

% subset_insertI2
tff(fact_2385_insert__subsetI,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,X6: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X6),A3))
       => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(insert8211810215607154385at_nat(X,X6)),A3)) ) ) ).

% insert_subsetI
tff(fact_2386_insert__subsetI,axiom,
    ! [X: real,A3: set_real,X6: set_real] :
      ( pp(aa_set_real_bool(member_real(X),A3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(X6),A3))
       => pp(aa_set_real_bool(ord_less_eq_set_real(insert_real(X,X6)),A3)) ) ) ).

% insert_subsetI
tff(fact_2387_insert__subsetI,axiom,
    ! [X: bool,A3: set_bool,X6: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(X),A3))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(X6),A3))
       => pp(aa_set_bool_bool(ord_less_eq_set_bool(insert_bool(X,X6)),A3)) ) ) ).

% insert_subsetI
tff(fact_2388_insert__subsetI,axiom,
    ! [X: set_nat,A3: set_set_nat,X6: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),A3))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(X6),A3))
       => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(insert_set_nat(X,X6)),A3)) ) ) ).

% insert_subsetI
tff(fact_2389_insert__subsetI,axiom,
    ! [X: nat,A3: set_nat,X6: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),A3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X6),A3))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,insert_nat(X,X6)),A3)) ) ) ).

% insert_subsetI
tff(fact_2390_insert__subsetI,axiom,
    ! [X: int,A3: set_int,X6: set_int] :
      ( pp(aa_set_int_bool(member_int(X),A3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X6),A3))
       => pp(aa_set_int_bool(ord_less_eq_set_int(insert_int(X,X6)),A3)) ) ) ).

% insert_subsetI
tff(fact_2391_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc9217583074051984497at_nat] :
      ~ ! [F3: fun_nat_fun_nat_nat,A: nat,B: nat,Acc: nat] : X != produc2818902965508115497at_nat(F3,produc487386426758144856at_nat(A,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_2392_insert__Diff__if,axiom,
    ! [X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X,A3)),B2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2) ) )
      & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X,A3)),B2) = insert8211810215607154385at_nat(X,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2)) ) ) ) ).

% insert_Diff_if
tff(fact_2393_insert__Diff__if,axiom,
    ! [X: real,B2: set_real,A3: set_real] :
      ( ( pp(aa_set_real_bool(member_real(X),B2))
       => ( minus_minus_set_real(insert_real(X,A3),B2) = minus_minus_set_real(A3,B2) ) )
      & ( ~ pp(aa_set_real_bool(member_real(X),B2))
       => ( minus_minus_set_real(insert_real(X,A3),B2) = insert_real(X,minus_minus_set_real(A3,B2)) ) ) ) ).

% insert_Diff_if
tff(fact_2394_insert__Diff__if,axiom,
    ! [X: bool,B2: set_bool,A3: set_bool] :
      ( ( pp(aa_set_bool_bool(member_bool(X),B2))
       => ( minus_minus_set_bool(insert_bool(X,A3),B2) = minus_minus_set_bool(A3,B2) ) )
      & ( ~ pp(aa_set_bool_bool(member_bool(X),B2))
       => ( minus_minus_set_bool(insert_bool(X,A3),B2) = insert_bool(X,minus_minus_set_bool(A3,B2)) ) ) ) ).

% insert_Diff_if
tff(fact_2395_insert__Diff__if,axiom,
    ! [X: set_nat,B2: set_set_nat,A3: set_set_nat] :
      ( ( pp(aa_set_set_nat_bool(member_set_nat(X),B2))
       => ( minus_2163939370556025621et_nat(insert_set_nat(X,A3),B2) = minus_2163939370556025621et_nat(A3,B2) ) )
      & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),B2))
       => ( minus_2163939370556025621et_nat(insert_set_nat(X,A3),B2) = insert_set_nat(X,minus_2163939370556025621et_nat(A3,B2)) ) ) ) ).

% insert_Diff_if
tff(fact_2396_insert__Diff__if,axiom,
    ! [X: int,B2: set_int,A3: set_int] :
      ( ( pp(aa_set_int_bool(member_int(X),B2))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,insert_int(X,A3)),B2) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2) ) )
      & ( ~ pp(aa_set_int_bool(member_int(X),B2))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,insert_int(X,A3)),B2) = insert_int(X,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2)) ) ) ) ).

% insert_Diff_if
tff(fact_2397_insert__Diff__if,axiom,
    ! [X: nat,B2: set_nat,A3: set_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(X),B2))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(X,A3)),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(X),B2))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(X,A3)),B2) = insert_nat(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)) ) ) ) ).

% insert_Diff_if
tff(fact_2398_finite_Ocases,axiom,
    ! [A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( A2 != bot_bot_set_complex )
       => ~ ! [A4: set_complex] :
              ( ? [A: complex] : A2 = insert_complex(A,A4)
             => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4)) ) ) ) ).

% finite.cases
tff(fact_2399_finite_Ocases,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => ( ( A2 != bot_bo2099793752762293965at_nat )
       => ~ ! [A4: set_Pr1261947904930325089at_nat] :
              ( ? [A: product_prod_nat_nat] : A2 = insert8211810215607154385at_nat(A,A4)
             => ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4)) ) ) ) ).

% finite.cases
tff(fact_2400_finite_Ocases,axiom,
    ! [A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ~ ! [A4: set_Extended_enat] :
              ( ? [A: extended_enat] : A2 = insert_Extended_enat(A,A4)
             => ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A4)) ) ) ) ).

% finite.cases
tff(fact_2401_finite_Ocases,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ~ ! [A4: set_real] :
              ( ? [A: real] : A2 = insert_real(A,A4)
             => ~ pp(aa_set_real_bool(finite_finite_real,A4)) ) ) ) ).

% finite.cases
tff(fact_2402_finite_Ocases,axiom,
    ! [A2: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( A2 != bot_bot_set_bool )
       => ~ ! [A4: set_bool] :
              ( ? [A: bool] : A2 = insert_bool(A,A4)
             => ~ pp(aa_set_bool_bool(finite_finite_bool,A4)) ) ) ) ).

% finite.cases
tff(fact_2403_finite_Ocases,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ~ ! [A4: set_nat] :
              ( ? [A: nat] : A2 = insert_nat(A,A4)
             => ~ pp(aa_set_nat_bool(finite_finite_nat,A4)) ) ) ) ).

% finite.cases
tff(fact_2404_finite_Ocases,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ~ ! [A4: set_int] :
              ( ? [A: int] : A2 = insert_int(A,A4)
             => ~ pp(aa_set_int_bool(finite_finite_int,A4)) ) ) ) ).

% finite.cases
tff(fact_2405_finite_Osimps,axiom,
    ! [A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
    <=> ( ( A2 = bot_bot_set_complex )
        | ? [A7: set_complex,A6: complex] :
            ( ( A2 = insert_complex(A6,A7) )
            & pp(aa_set_complex_bool(finite3207457112153483333omplex,A7)) ) ) ) ).

% finite.simps
tff(fact_2406_finite_Osimps,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
    <=> ( ( A2 = bot_bo2099793752762293965at_nat )
        | ? [A7: set_Pr1261947904930325089at_nat,A6: product_prod_nat_nat] :
            ( ( A2 = insert8211810215607154385at_nat(A6,A7) )
            & pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A7)) ) ) ) ).

% finite.simps
tff(fact_2407_finite_Osimps,axiom,
    ! [A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
    <=> ( ( A2 = bot_bo7653980558646680370d_enat )
        | ? [A7: set_Extended_enat,A6: extended_enat] :
            ( ( A2 = insert_Extended_enat(A6,A7) )
            & pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A7)) ) ) ) ).

% finite.simps
tff(fact_2408_finite_Osimps,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
    <=> ( ( A2 = bot_bot_set_real )
        | ? [A7: set_real,A6: real] :
            ( ( A2 = insert_real(A6,A7) )
            & pp(aa_set_real_bool(finite_finite_real,A7)) ) ) ) ).

% finite.simps
tff(fact_2409_finite_Osimps,axiom,
    ! [A2: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
    <=> ( ( A2 = bot_bot_set_bool )
        | ? [A7: set_bool,A6: bool] :
            ( ( A2 = insert_bool(A6,A7) )
            & pp(aa_set_bool_bool(finite_finite_bool,A7)) ) ) ) ).

% finite.simps
tff(fact_2410_finite_Osimps,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
    <=> ( ( A2 = bot_bot_set_nat )
        | ? [A7: set_nat,A6: nat] :
            ( ( A2 = insert_nat(A6,A7) )
            & pp(aa_set_nat_bool(finite_finite_nat,A7)) ) ) ) ).

% finite.simps
tff(fact_2411_finite_Osimps,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
    <=> ( ( A2 = bot_bot_set_int )
        | ? [A7: set_int,A6: int] :
            ( ( A2 = insert_int(A6,A7) )
            & pp(aa_set_int_bool(finite_finite_int,A7)) ) ) ) ).

% finite.simps
tff(fact_2412_finite__induct,axiom,
    ! [F2: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F2))
     => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
       => ( ! [X5: set_nat,F4: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
             => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X5),F4))
               => ( pp(aa_set_set_nat_bool(P,F4))
                 => pp(aa_set_set_nat_bool(P,insert_set_nat(X5,F4))) ) ) )
         => pp(aa_set_set_nat_bool(P,F2)) ) ) ) ).

% finite_induct
tff(fact_2413_finite__induct,axiom,
    ! [F2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F2))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X5: complex,F4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
             => ( ~ pp(aa_set_complex_bool(member_complex(X5),F4))
               => ( pp(aa_set_complex_bool(P,F4))
                 => pp(aa_set_complex_bool(P,insert_complex(X5,F4))) ) ) )
         => pp(aa_set_complex_bool(P,F2)) ) ) ) ).

% finite_induct
tff(fact_2414_finite__induct,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F2))
     => ( pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat))
       => ( ! [X5: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F4))
             => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),F4))
               => ( pp(aa_set3135850010578026034t_bool(P,F4))
                 => pp(aa_set3135850010578026034t_bool(P,insert8211810215607154385at_nat(X5,F4))) ) ) )
         => pp(aa_set3135850010578026034t_bool(P,F2)) ) ) ) ).

% finite_induct
tff(fact_2415_finite__induct,axiom,
    ! [F2: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F2))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [X5: extended_enat,F4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F4))
             => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),F4))
               => ( pp(aa_set3347977610044113549t_bool(P,F4))
                 => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(X5,F4))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,F2)) ) ) ) ).

% finite_induct
tff(fact_2416_finite__induct,axiom,
    ! [F2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X5: real,F4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,F4))
             => ( ~ pp(aa_set_real_bool(member_real(X5),F4))
               => ( pp(aa_set_real_bool(P,F4))
                 => pp(aa_set_real_bool(P,insert_real(X5,F4))) ) ) )
         => pp(aa_set_real_bool(P,F2)) ) ) ) ).

% finite_induct
tff(fact_2417_finite__induct,axiom,
    ! [F2: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,F2))
     => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
       => ( ! [X5: bool,F4: set_bool] :
              ( pp(aa_set_bool_bool(finite_finite_bool,F4))
             => ( ~ pp(aa_set_bool_bool(member_bool(X5),F4))
               => ( pp(aa_set_bool_bool(P,F4))
                 => pp(aa_set_bool_bool(P,insert_bool(X5,F4))) ) ) )
         => pp(aa_set_bool_bool(P,F2)) ) ) ) ).

% finite_induct
tff(fact_2418_finite__induct,axiom,
    ! [F2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X5: nat,F4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,F4))
             => ( ~ pp(aa_set_nat_bool(member_nat(X5),F4))
               => ( pp(aa_set_nat_bool(P,F4))
                 => pp(aa_set_nat_bool(P,insert_nat(X5,F4))) ) ) )
         => pp(aa_set_nat_bool(P,F2)) ) ) ) ).

% finite_induct
tff(fact_2419_finite__induct,axiom,
    ! [F2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X5: int,F4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,F4))
             => ( ~ pp(aa_set_int_bool(member_int(X5),F4))
               => ( pp(aa_set_int_bool(P,F4))
                 => pp(aa_set_int_bool(P,insert_int(X5,F4))) ) ) )
         => pp(aa_set_int_bool(P,F2)) ) ) ) ).

% finite_induct
tff(fact_2420_finite__ne__induct,axiom,
    ! [F2: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F2))
     => ( ( F2 != bot_bot_set_set_nat )
       => ( ! [X5: set_nat] : pp(aa_set_set_nat_bool(P,insert_set_nat(X5,bot_bot_set_set_nat)))
         => ( ! [X5: set_nat,F4: set_set_nat] :
                ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
               => ( ( F4 != bot_bot_set_set_nat )
                 => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X5),F4))
                   => ( pp(aa_set_set_nat_bool(P,F4))
                     => pp(aa_set_set_nat_bool(P,insert_set_nat(X5,F4))) ) ) ) )
           => pp(aa_set_set_nat_bool(P,F2)) ) ) ) ) ).

% finite_ne_induct
tff(fact_2421_finite__ne__induct,axiom,
    ! [F2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F2))
     => ( ( F2 != bot_bot_set_complex )
       => ( ! [X5: complex] : pp(aa_set_complex_bool(P,insert_complex(X5,bot_bot_set_complex)))
         => ( ! [X5: complex,F4: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
               => ( ( F4 != bot_bot_set_complex )
                 => ( ~ pp(aa_set_complex_bool(member_complex(X5),F4))
                   => ( pp(aa_set_complex_bool(P,F4))
                     => pp(aa_set_complex_bool(P,insert_complex(X5,F4))) ) ) ) )
           => pp(aa_set_complex_bool(P,F2)) ) ) ) ) ).

% finite_ne_induct
tff(fact_2422_finite__ne__induct,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F2))
     => ( ( F2 != bot_bo2099793752762293965at_nat )
       => ( ! [X5: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(P,insert8211810215607154385at_nat(X5,bot_bo2099793752762293965at_nat)))
         => ( ! [X5: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F4))
               => ( ( F4 != bot_bo2099793752762293965at_nat )
                 => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),F4))
                   => ( pp(aa_set3135850010578026034t_bool(P,F4))
                     => pp(aa_set3135850010578026034t_bool(P,insert8211810215607154385at_nat(X5,F4))) ) ) ) )
           => pp(aa_set3135850010578026034t_bool(P,F2)) ) ) ) ) ).

% finite_ne_induct
tff(fact_2423_finite__ne__induct,axiom,
    ! [F2: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F2))
     => ( ( F2 != bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat] : pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(X5,bot_bo7653980558646680370d_enat)))
         => ( ! [X5: extended_enat,F4: set_Extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F4))
               => ( ( F4 != bot_bo7653980558646680370d_enat )
                 => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),F4))
                   => ( pp(aa_set3347977610044113549t_bool(P,F4))
                     => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(X5,F4))) ) ) ) )
           => pp(aa_set3347977610044113549t_bool(P,F2)) ) ) ) ) ).

% finite_ne_induct
tff(fact_2424_finite__ne__induct,axiom,
    ! [F2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F2))
     => ( ( F2 != bot_bot_set_real )
       => ( ! [X5: real] : pp(aa_set_real_bool(P,insert_real(X5,bot_bot_set_real)))
         => ( ! [X5: real,F4: set_real] :
                ( pp(aa_set_real_bool(finite_finite_real,F4))
               => ( ( F4 != bot_bot_set_real )
                 => ( ~ pp(aa_set_real_bool(member_real(X5),F4))
                   => ( pp(aa_set_real_bool(P,F4))
                     => pp(aa_set_real_bool(P,insert_real(X5,F4))) ) ) ) )
           => pp(aa_set_real_bool(P,F2)) ) ) ) ) ).

% finite_ne_induct
tff(fact_2425_finite__ne__induct,axiom,
    ! [F2: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,F2))
     => ( ( F2 != bot_bot_set_bool )
       => ( ! [X5: bool] : pp(aa_set_bool_bool(P,insert_bool(X5,bot_bot_set_bool)))
         => ( ! [X5: bool,F4: set_bool] :
                ( pp(aa_set_bool_bool(finite_finite_bool,F4))
               => ( ( F4 != bot_bot_set_bool )
                 => ( ~ pp(aa_set_bool_bool(member_bool(X5),F4))
                   => ( pp(aa_set_bool_bool(P,F4))
                     => pp(aa_set_bool_bool(P,insert_bool(X5,F4))) ) ) ) )
           => pp(aa_set_bool_bool(P,F2)) ) ) ) ) ).

% finite_ne_induct
tff(fact_2426_finite__ne__induct,axiom,
    ! [F2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F2))
     => ( ( F2 != bot_bot_set_nat )
       => ( ! [X5: nat] : pp(aa_set_nat_bool(P,insert_nat(X5,bot_bot_set_nat)))
         => ( ! [X5: nat,F4: set_nat] :
                ( pp(aa_set_nat_bool(finite_finite_nat,F4))
               => ( ( F4 != bot_bot_set_nat )
                 => ( ~ pp(aa_set_nat_bool(member_nat(X5),F4))
                   => ( pp(aa_set_nat_bool(P,F4))
                     => pp(aa_set_nat_bool(P,insert_nat(X5,F4))) ) ) ) )
           => pp(aa_set_nat_bool(P,F2)) ) ) ) ) ).

% finite_ne_induct
tff(fact_2427_finite__ne__induct,axiom,
    ! [F2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F2))
     => ( ( F2 != bot_bot_set_int )
       => ( ! [X5: int] : pp(aa_set_int_bool(P,insert_int(X5,bot_bot_set_int)))
         => ( ! [X5: int,F4: set_int] :
                ( pp(aa_set_int_bool(finite_finite_int,F4))
               => ( ( F4 != bot_bot_set_int )
                 => ( ~ pp(aa_set_int_bool(member_int(X5),F4))
                   => ( pp(aa_set_int_bool(P,F4))
                     => pp(aa_set_int_bool(P,insert_int(X5,F4))) ) ) ) )
           => pp(aa_set_int_bool(P,F2)) ) ) ) ) ).

% finite_ne_induct
tff(fact_2428_infinite__finite__induct,axiom,
    ! [P: fun_set_set_nat_bool,A3: set_set_nat] :
      ( ! [A4: set_set_nat] :
          ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
         => pp(aa_set_set_nat_bool(P,A4)) )
     => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
       => ( ! [X5: set_nat,F4: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
             => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X5),F4))
               => ( pp(aa_set_set_nat_bool(P,F4))
                 => pp(aa_set_set_nat_bool(P,insert_set_nat(X5,F4))) ) ) )
         => pp(aa_set_set_nat_bool(P,A3)) ) ) ) ).

% infinite_finite_induct
tff(fact_2429_infinite__finite__induct,axiom,
    ! [P: fun_set_complex_bool,A3: set_complex] :
      ( ! [A4: set_complex] :
          ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
         => pp(aa_set_complex_bool(P,A4)) )
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X5: complex,F4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
             => ( ~ pp(aa_set_complex_bool(member_complex(X5),F4))
               => ( pp(aa_set_complex_bool(P,F4))
                 => pp(aa_set_complex_bool(P,insert_complex(X5,F4))) ) ) )
         => pp(aa_set_complex_bool(P,A3)) ) ) ) ).

% infinite_finite_induct
tff(fact_2430_infinite__finite__induct,axiom,
    ! [P: fun_se8074476271188067385t_bool,A3: set_Pr1261947904930325089at_nat] :
      ( ! [A4: set_Pr1261947904930325089at_nat] :
          ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
         => pp(aa_set3135850010578026034t_bool(P,A4)) )
     => ( pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat))
       => ( ! [X5: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F4))
             => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),F4))
               => ( pp(aa_set3135850010578026034t_bool(P,F4))
                 => pp(aa_set3135850010578026034t_bool(P,insert8211810215607154385at_nat(X5,F4))) ) ) )
         => pp(aa_set3135850010578026034t_bool(P,A3)) ) ) ) ).

% infinite_finite_induct
tff(fact_2431_infinite__finite__induct,axiom,
    ! [P: fun_se2015444840911768724t_bool,A3: set_Extended_enat] :
      ( ! [A4: set_Extended_enat] :
          ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A4))
         => pp(aa_set3347977610044113549t_bool(P,A4)) )
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [X5: extended_enat,F4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F4))
             => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),F4))
               => ( pp(aa_set3347977610044113549t_bool(P,F4))
                 => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(X5,F4))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,A3)) ) ) ) ).

% infinite_finite_induct
tff(fact_2432_infinite__finite__induct,axiom,
    ! [P: fun_set_real_bool,A3: set_real] :
      ( ! [A4: set_real] :
          ( ~ pp(aa_set_real_bool(finite_finite_real,A4))
         => pp(aa_set_real_bool(P,A4)) )
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X5: real,F4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,F4))
             => ( ~ pp(aa_set_real_bool(member_real(X5),F4))
               => ( pp(aa_set_real_bool(P,F4))
                 => pp(aa_set_real_bool(P,insert_real(X5,F4))) ) ) )
         => pp(aa_set_real_bool(P,A3)) ) ) ) ).

% infinite_finite_induct
tff(fact_2433_infinite__finite__induct,axiom,
    ! [P: fun_set_bool_bool,A3: set_bool] :
      ( ! [A4: set_bool] :
          ( ~ pp(aa_set_bool_bool(finite_finite_bool,A4))
         => pp(aa_set_bool_bool(P,A4)) )
     => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
       => ( ! [X5: bool,F4: set_bool] :
              ( pp(aa_set_bool_bool(finite_finite_bool,F4))
             => ( ~ pp(aa_set_bool_bool(member_bool(X5),F4))
               => ( pp(aa_set_bool_bool(P,F4))
                 => pp(aa_set_bool_bool(P,insert_bool(X5,F4))) ) ) )
         => pp(aa_set_bool_bool(P,A3)) ) ) ) ).

% infinite_finite_induct
tff(fact_2434_infinite__finite__induct,axiom,
    ! [P: fun_set_nat_bool,A3: set_nat] :
      ( ! [A4: set_nat] :
          ( ~ pp(aa_set_nat_bool(finite_finite_nat,A4))
         => pp(aa_set_nat_bool(P,A4)) )
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X5: nat,F4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,F4))
             => ( ~ pp(aa_set_nat_bool(member_nat(X5),F4))
               => ( pp(aa_set_nat_bool(P,F4))
                 => pp(aa_set_nat_bool(P,insert_nat(X5,F4))) ) ) )
         => pp(aa_set_nat_bool(P,A3)) ) ) ) ).

% infinite_finite_induct
tff(fact_2435_infinite__finite__induct,axiom,
    ! [P: fun_set_int_bool,A3: set_int] :
      ( ! [A4: set_int] :
          ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
         => pp(aa_set_int_bool(P,A4)) )
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X5: int,F4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,F4))
             => ( ~ pp(aa_set_int_bool(member_int(X5),F4))
               => ( pp(aa_set_int_bool(P,F4))
                 => pp(aa_set_int_bool(P,insert_int(X5,F4))) ) ) )
         => pp(aa_set_int_bool(P,A3)) ) ) ) ).

% infinite_finite_induct
tff(fact_2436_subset__singletonD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)))
     => ( ( A3 = bot_bo2099793752762293965at_nat )
        | ( A3 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ).

% subset_singletonD
tff(fact_2437_subset__singletonD,axiom,
    ! [A3: set_real,X: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),insert_real(X,bot_bot_set_real)))
     => ( ( A3 = bot_bot_set_real )
        | ( A3 = insert_real(X,bot_bot_set_real) ) ) ) ).

% subset_singletonD
tff(fact_2438_subset__singletonD,axiom,
    ! [A3: set_bool,X: bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),insert_bool(X,bot_bot_set_bool)))
     => ( ( A3 = bot_bot_set_bool )
        | ( A3 = insert_bool(X,bot_bot_set_bool) ) ) ) ).

% subset_singletonD
tff(fact_2439_subset__singletonD,axiom,
    ! [A3: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),insert_nat(X,bot_bot_set_nat)))
     => ( ( A3 = bot_bot_set_nat )
        | ( A3 = insert_nat(X,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_2440_subset__singletonD,axiom,
    ! [A3: set_int,X: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),insert_int(X,bot_bot_set_int)))
     => ( ( A3 = bot_bot_set_int )
        | ( A3 = insert_int(X,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_2441_subset__singleton__iff,axiom,
    ! [X6: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X6),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat)))
    <=> ( ( X6 = bot_bo2099793752762293965at_nat )
        | ( X6 = insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_2442_subset__singleton__iff,axiom,
    ! [X6: set_real,A2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(X6),insert_real(A2,bot_bot_set_real)))
    <=> ( ( X6 = bot_bot_set_real )
        | ( X6 = insert_real(A2,bot_bot_set_real) ) ) ) ).

% subset_singleton_iff
tff(fact_2443_subset__singleton__iff,axiom,
    ! [X6: set_bool,A2: bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(X6),insert_bool(A2,bot_bot_set_bool)))
    <=> ( ( X6 = bot_bot_set_bool )
        | ( X6 = insert_bool(A2,bot_bot_set_bool) ) ) ) ).

% subset_singleton_iff
tff(fact_2444_subset__singleton__iff,axiom,
    ! [X6: set_nat,A2: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X6),insert_nat(A2,bot_bot_set_nat)))
    <=> ( ( X6 = bot_bot_set_nat )
        | ( X6 = insert_nat(A2,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_2445_subset__singleton__iff,axiom,
    ! [X6: set_int,A2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X6),insert_int(A2,bot_bot_set_int)))
    <=> ( ( X6 = bot_bot_set_int )
        | ( X6 = insert_int(A2,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_2446_Diff__insert,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(A2,B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2)),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat)) ).

% Diff_insert
tff(fact_2447_Diff__insert,axiom,
    ! [A3: set_real,A2: real,B2: set_real] : minus_minus_set_real(A3,insert_real(A2,B2)) = minus_minus_set_real(minus_minus_set_real(A3,B2),insert_real(A2,bot_bot_set_real)) ).

% Diff_insert
tff(fact_2448_Diff__insert,axiom,
    ! [A3: set_bool,A2: bool,B2: set_bool] : minus_minus_set_bool(A3,insert_bool(A2,B2)) = minus_minus_set_bool(minus_minus_set_bool(A3,B2),insert_bool(A2,bot_bot_set_bool)) ).

% Diff_insert
tff(fact_2449_Diff__insert,axiom,
    ! [A3: set_int,A2: int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(A2,B2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2)),insert_int(A2,bot_bot_set_int)) ).

% Diff_insert
tff(fact_2450_Diff__insert,axiom,
    ! [A3: set_nat,A2: nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(A2,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)),insert_nat(A2,bot_bot_set_nat)) ).

% Diff_insert
tff(fact_2451_insert__Diff,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A3))
     => ( insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat))) = A3 ) ) ).

% insert_Diff
tff(fact_2452_insert__Diff,axiom,
    ! [A2: set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A2),A3))
     => ( insert_set_nat(A2,minus_2163939370556025621et_nat(A3,insert_set_nat(A2,bot_bot_set_set_nat))) = A3 ) ) ).

% insert_Diff
tff(fact_2453_insert__Diff,axiom,
    ! [A2: real,A3: set_real] :
      ( pp(aa_set_real_bool(member_real(A2),A3))
     => ( insert_real(A2,minus_minus_set_real(A3,insert_real(A2,bot_bot_set_real))) = A3 ) ) ).

% insert_Diff
tff(fact_2454_insert__Diff,axiom,
    ! [A2: bool,A3: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(A2),A3))
     => ( insert_bool(A2,minus_minus_set_bool(A3,insert_bool(A2,bot_bot_set_bool))) = A3 ) ) ).

% insert_Diff
tff(fact_2455_insert__Diff,axiom,
    ! [A2: int,A3: set_int] :
      ( pp(aa_set_int_bool(member_int(A2),A3))
     => ( insert_int(A2,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(A2,bot_bot_set_int))) = A3 ) ) ).

% insert_Diff
tff(fact_2456_insert__Diff,axiom,
    ! [A2: nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A2),A3))
     => ( insert_nat(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(A2,bot_bot_set_nat))) = A3 ) ) ).

% insert_Diff
tff(fact_2457_Diff__insert2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(A2,B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat))),B2) ).

% Diff_insert2
tff(fact_2458_Diff__insert2,axiom,
    ! [A3: set_real,A2: real,B2: set_real] : minus_minus_set_real(A3,insert_real(A2,B2)) = minus_minus_set_real(minus_minus_set_real(A3,insert_real(A2,bot_bot_set_real)),B2) ).

% Diff_insert2
tff(fact_2459_Diff__insert2,axiom,
    ! [A3: set_bool,A2: bool,B2: set_bool] : minus_minus_set_bool(A3,insert_bool(A2,B2)) = minus_minus_set_bool(minus_minus_set_bool(A3,insert_bool(A2,bot_bot_set_bool)),B2) ).

% Diff_insert2
tff(fact_2460_Diff__insert2,axiom,
    ! [A3: set_int,A2: int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(A2,B2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(A2,bot_bot_set_int))),B2) ).

% Diff_insert2
tff(fact_2461_Diff__insert2,axiom,
    ! [A3: set_nat,A2: nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(A2,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(A2,bot_bot_set_nat))),B2) ).

% Diff_insert2
tff(fact_2462_Diff__insert__absorb,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X,A3)),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) = A3 ) ) ).

% Diff_insert_absorb
tff(fact_2463_Diff__insert__absorb,axiom,
    ! [X: set_nat,A3: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A3))
     => ( minus_2163939370556025621et_nat(insert_set_nat(X,A3),insert_set_nat(X,bot_bot_set_set_nat)) = A3 ) ) ).

% Diff_insert_absorb
tff(fact_2464_Diff__insert__absorb,axiom,
    ! [X: real,A3: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X),A3))
     => ( minus_minus_set_real(insert_real(X,A3),insert_real(X,bot_bot_set_real)) = A3 ) ) ).

% Diff_insert_absorb
tff(fact_2465_Diff__insert__absorb,axiom,
    ! [X: bool,A3: set_bool] :
      ( ~ pp(aa_set_bool_bool(member_bool(X),A3))
     => ( minus_minus_set_bool(insert_bool(X,A3),insert_bool(X,bot_bot_set_bool)) = A3 ) ) ).

% Diff_insert_absorb
tff(fact_2466_Diff__insert__absorb,axiom,
    ! [X: int,A3: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X),A3))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,insert_int(X,A3)),insert_int(X,bot_bot_set_int)) = A3 ) ) ).

% Diff_insert_absorb
tff(fact_2467_Diff__insert__absorb,axiom,
    ! [X: nat,A3: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),A3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(X,A3)),insert_nat(X,bot_bot_set_nat)) = A3 ) ) ).

% Diff_insert_absorb
tff(fact_2468_subset__Diff__insert,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),insert8211810215607154385at_nat(X,C2))))
    <=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),C2)))
        & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3)) ) ) ).

% subset_Diff_insert
tff(fact_2469_subset__Diff__insert,axiom,
    ! [A3: set_real,B2: set_real,X: real,C2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),minus_minus_set_real(B2,insert_real(X,C2))))
    <=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),minus_minus_set_real(B2,C2)))
        & ~ pp(aa_set_real_bool(member_real(X),A3)) ) ) ).

% subset_Diff_insert
tff(fact_2470_subset__Diff__insert,axiom,
    ! [A3: set_bool,B2: set_bool,X: bool,C2: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),minus_minus_set_bool(B2,insert_bool(X,C2))))
    <=> ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),minus_minus_set_bool(B2,C2)))
        & ~ pp(aa_set_bool_bool(member_bool(X),A3)) ) ) ).

% subset_Diff_insert
tff(fact_2471_subset__Diff__insert,axiom,
    ! [A3: set_set_nat,B2: set_set_nat,X: set_nat,C2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),minus_2163939370556025621et_nat(B2,insert_set_nat(X,C2))))
    <=> ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),minus_2163939370556025621et_nat(B2,C2)))
        & ~ pp(aa_set_set_nat_bool(member_set_nat(X),A3)) ) ) ).

% subset_Diff_insert
tff(fact_2472_subset__Diff__insert,axiom,
    ! [A3: set_nat,B2: set_nat,X: nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),insert_nat(X,C2))))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),C2)))
        & ~ pp(aa_set_nat_bool(member_nat(X),A3)) ) ) ).

% subset_Diff_insert
tff(fact_2473_subset__Diff__insert,axiom,
    ! [A3: set_int,B2: set_int,X: int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B2),insert_int(X,C2))))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B2),C2)))
        & ~ pp(aa_set_int_bool(member_int(X),A3)) ) ) ).

% subset_Diff_insert
tff(fact_2474_card__insert__le,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,insert8211810215607154385at_nat(X,A3)))) ).

% card_insert_le
tff(fact_2475_card__insert__le,axiom,
    ! [A3: set_real,X: real] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(A3)),finite_card_real(insert_real(X,A3)))) ).

% card_insert_le
tff(fact_2476_card__insert__le,axiom,
    ! [A3: set_bool,X: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_bool(A3)),finite_card_bool(insert_bool(X,A3)))) ).

% card_insert_le
tff(fact_2477_card__insert__le,axiom,
    ! [A3: set_complex,X: complex] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(A3)),finite_card_complex(insert_complex(X,A3)))) ).

% card_insert_le
tff(fact_2478_card__insert__le,axiom,
    ! [A3: set_list_nat,X: list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(A3)),finite_card_list_nat(insert_list_nat(X,A3)))) ).

% card_insert_le
tff(fact_2479_card__insert__le,axiom,
    ! [A3: set_set_nat,X: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(A3)),finite_card_set_nat(insert_set_nat(X,A3)))) ).

% card_insert_le
tff(fact_2480_card__insert__le,axiom,
    ! [A3: set_nat,X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,insert_nat(X,A3)))) ).

% card_insert_le
tff(fact_2481_card__insert__le,axiom,
    ! [A3: set_int,X: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(A3)),finite_card_int(insert_int(X,A3)))) ).

% card_insert_le
tff(fact_2482_finite__ranking__induct,axiom,
    ! [S: set_complex,P: fun_set_complex_bool,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X5: complex,S4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S4))
             => ( ! [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),S4))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,Y4)),aa_complex_rat(F,X5))) )
               => ( pp(aa_set_complex_bool(P,S4))
                 => pp(aa_set_complex_bool(P,insert_complex(X5,S4))) ) ) )
         => pp(aa_set_complex_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_2483_finite__ranking__induct,axiom,
    ! [S: set_Extended_enat,P: fun_se2015444840911768724t_bool,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [X5: extended_enat,S4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S4))
             => ( ! [Y4: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y4),S4))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Extended_enat_rat(F,Y4)),aa_Extended_enat_rat(F,X5))) )
               => ( pp(aa_set3347977610044113549t_bool(P,S4))
                 => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(X5,S4))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_2484_finite__ranking__induct,axiom,
    ! [S: set_real,P: fun_set_real_bool,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X5: real,S4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S4))
             => ( ! [Y4: real] :
                    ( pp(aa_set_real_bool(member_real(Y4),S4))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,Y4)),aa_real_rat(F,X5))) )
               => ( pp(aa_set_real_bool(P,S4))
                 => pp(aa_set_real_bool(P,insert_real(X5,S4))) ) ) )
         => pp(aa_set_real_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_2485_finite__ranking__induct,axiom,
    ! [S: set_bool,P: fun_set_bool_bool,F: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
       => ( ! [X5: bool,S4: set_bool] :
              ( pp(aa_set_bool_bool(finite_finite_bool,S4))
             => ( ! [Y4: bool] :
                    ( pp(aa_set_bool_bool(member_bool(Y4),S4))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_bool_rat(F,Y4)),aa_bool_rat(F,X5))) )
               => ( pp(aa_set_bool_bool(P,S4))
                 => pp(aa_set_bool_bool(P,insert_bool(X5,S4))) ) ) )
         => pp(aa_set_bool_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_2486_finite__ranking__induct,axiom,
    ! [S: set_nat,P: fun_set_nat_bool,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X5: nat,S4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,S4))
             => ( ! [Y4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y4),S4))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,Y4)),aa_nat_rat(F,X5))) )
               => ( pp(aa_set_nat_bool(P,S4))
                 => pp(aa_set_nat_bool(P,insert_nat(X5,S4))) ) ) )
         => pp(aa_set_nat_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_2487_finite__ranking__induct,axiom,
    ! [S: set_int,P: fun_set_int_bool,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X5: int,S4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,S4))
             => ( ! [Y4: int] :
                    ( pp(aa_set_int_bool(member_int(Y4),S4))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,Y4)),aa_int_rat(F,X5))) )
               => ( pp(aa_set_int_bool(P,S4))
                 => pp(aa_set_int_bool(P,insert_int(X5,S4))) ) ) )
         => pp(aa_set_int_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_2488_finite__ranking__induct,axiom,
    ! [S: set_complex,P: fun_set_complex_bool,F: fun_complex_num] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X5: complex,S4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S4))
             => ( ! [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),S4))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_complex_num(F,Y4)),aa_complex_num(F,X5))) )
               => ( pp(aa_set_complex_bool(P,S4))
                 => pp(aa_set_complex_bool(P,insert_complex(X5,S4))) ) ) )
         => pp(aa_set_complex_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_2489_finite__ranking__induct,axiom,
    ! [S: set_Extended_enat,P: fun_se2015444840911768724t_bool,F: fun_Ex8368542335153306515at_num] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [X5: extended_enat,S4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S4))
             => ( ! [Y4: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y4),S4))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_Extended_enat_num(F,Y4)),aa_Extended_enat_num(F,X5))) )
               => ( pp(aa_set3347977610044113549t_bool(P,S4))
                 => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(X5,S4))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_2490_finite__ranking__induct,axiom,
    ! [S: set_real,P: fun_set_real_bool,F: fun_real_num] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X5: real,S4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S4))
             => ( ! [Y4: real] :
                    ( pp(aa_set_real_bool(member_real(Y4),S4))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_real_num(F,Y4)),aa_real_num(F,X5))) )
               => ( pp(aa_set_real_bool(P,S4))
                 => pp(aa_set_real_bool(P,insert_real(X5,S4))) ) ) )
         => pp(aa_set_real_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_2491_finite__ranking__induct,axiom,
    ! [S: set_bool,P: fun_set_bool_bool,F: fun_bool_num] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
       => ( ! [X5: bool,S4: set_bool] :
              ( pp(aa_set_bool_bool(finite_finite_bool,S4))
             => ( ! [Y4: bool] :
                    ( pp(aa_set_bool_bool(member_bool(Y4),S4))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_bool_num(F,Y4)),aa_bool_num(F,X5))) )
               => ( pp(aa_set_bool_bool(P,S4))
                 => pp(aa_set_bool_bool(P,insert_bool(X5,S4))) ) ) )
         => pp(aa_set_bool_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_2492_finite__linorder__max__induct,axiom,
    ! [A3: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [B: extended_enat,A4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A4))
             => ( ! [X4: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A4))
                   => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,X4),B)) )
               => ( pp(aa_set3347977610044113549t_bool(P,A4))
                 => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(B,A4))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,A3)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_2493_finite__linorder__max__induct,axiom,
    ! [A3: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
       => ( ! [B: bool,A4: set_bool] :
              ( pp(aa_set_bool_bool(finite_finite_bool,A4))
             => ( ! [X4: bool] :
                    ( pp(aa_set_bool_bool(member_bool(X4),A4))
                   => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,X4),B)) )
               => ( pp(aa_set_bool_bool(P,A4))
                 => pp(aa_set_bool_bool(P,insert_bool(B,A4))) ) ) )
         => pp(aa_set_bool_bool(P,A3)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_2494_finite__linorder__max__induct,axiom,
    ! [A3: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [B: real,A4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A4))
             => ( ! [X4: real] :
                    ( pp(aa_set_real_bool(member_real(X4),A4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B)) )
               => ( pp(aa_set_real_bool(P,A4))
                 => pp(aa_set_real_bool(P,insert_real(B,A4))) ) ) )
         => pp(aa_set_real_bool(P,A3)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_2495_finite__linorder__max__induct,axiom,
    ! [A3: set_rat,P: fun_set_rat_bool] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A3))
     => ( pp(aa_set_rat_bool(P,bot_bot_set_rat))
       => ( ! [B: rat,A4: set_rat] :
              ( pp(aa_set_rat_bool(finite_finite_rat,A4))
             => ( ! [X4: rat] :
                    ( pp(aa_set_rat_bool(member_rat(X4),A4))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),B)) )
               => ( pp(aa_set_rat_bool(P,A4))
                 => pp(aa_set_rat_bool(P,insert_rat(B,A4))) ) ) )
         => pp(aa_set_rat_bool(P,A3)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_2496_finite__linorder__max__induct,axiom,
    ! [A3: set_num,P: fun_set_num_bool] :
      ( pp(aa_set_num_bool(finite_finite_num,A3))
     => ( pp(aa_set_num_bool(P,bot_bot_set_num))
       => ( ! [B: num,A4: set_num] :
              ( pp(aa_set_num_bool(finite_finite_num,A4))
             => ( ! [X4: num] :
                    ( pp(aa_set_num_bool(member_num(X4),A4))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),B)) )
               => ( pp(aa_set_num_bool(P,A4))
                 => pp(aa_set_num_bool(P,insert_num(B,A4))) ) ) )
         => pp(aa_set_num_bool(P,A3)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_2497_finite__linorder__max__induct,axiom,
    ! [A3: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [B: nat,A4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A4))
             => ( ! [X4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X4),A4))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),B)) )
               => ( pp(aa_set_nat_bool(P,A4))
                 => pp(aa_set_nat_bool(P,insert_nat(B,A4))) ) ) )
         => pp(aa_set_nat_bool(P,A3)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_2498_finite__linorder__max__induct,axiom,
    ! [A3: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [B: int,A4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A4))
             => ( ! [X4: int] :
                    ( pp(aa_set_int_bool(member_int(X4),A4))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),B)) )
               => ( pp(aa_set_int_bool(P,A4))
                 => pp(aa_set_int_bool(P,insert_int(B,A4))) ) ) )
         => pp(aa_set_int_bool(P,A3)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_2499_finite__linorder__min__induct,axiom,
    ! [A3: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [B: extended_enat,A4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A4))
             => ( ! [X4: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A4))
                   => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B),X4)) )
               => ( pp(aa_set3347977610044113549t_bool(P,A4))
                 => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(B,A4))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,A3)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_2500_finite__linorder__min__induct,axiom,
    ! [A3: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
       => ( ! [B: bool,A4: set_bool] :
              ( pp(aa_set_bool_bool(finite_finite_bool,A4))
             => ( ! [X4: bool] :
                    ( pp(aa_set_bool_bool(member_bool(X4),A4))
                   => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,B),X4)) )
               => ( pp(aa_set_bool_bool(P,A4))
                 => pp(aa_set_bool_bool(P,insert_bool(B,A4))) ) ) )
         => pp(aa_set_bool_bool(P,A3)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_2501_finite__linorder__min__induct,axiom,
    ! [A3: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [B: real,A4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A4))
             => ( ! [X4: real] :
                    ( pp(aa_set_real_bool(member_real(X4),A4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),X4)) )
               => ( pp(aa_set_real_bool(P,A4))
                 => pp(aa_set_real_bool(P,insert_real(B,A4))) ) ) )
         => pp(aa_set_real_bool(P,A3)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_2502_finite__linorder__min__induct,axiom,
    ! [A3: set_rat,P: fun_set_rat_bool] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A3))
     => ( pp(aa_set_rat_bool(P,bot_bot_set_rat))
       => ( ! [B: rat,A4: set_rat] :
              ( pp(aa_set_rat_bool(finite_finite_rat,A4))
             => ( ! [X4: rat] :
                    ( pp(aa_set_rat_bool(member_rat(X4),A4))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),X4)) )
               => ( pp(aa_set_rat_bool(P,A4))
                 => pp(aa_set_rat_bool(P,insert_rat(B,A4))) ) ) )
         => pp(aa_set_rat_bool(P,A3)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_2503_finite__linorder__min__induct,axiom,
    ! [A3: set_num,P: fun_set_num_bool] :
      ( pp(aa_set_num_bool(finite_finite_num,A3))
     => ( pp(aa_set_num_bool(P,bot_bot_set_num))
       => ( ! [B: num,A4: set_num] :
              ( pp(aa_set_num_bool(finite_finite_num,A4))
             => ( ! [X4: num] :
                    ( pp(aa_set_num_bool(member_num(X4),A4))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),X4)) )
               => ( pp(aa_set_num_bool(P,A4))
                 => pp(aa_set_num_bool(P,insert_num(B,A4))) ) ) )
         => pp(aa_set_num_bool(P,A3)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_2504_finite__linorder__min__induct,axiom,
    ! [A3: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [B: nat,A4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A4))
             => ( ! [X4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X4),A4))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),X4)) )
               => ( pp(aa_set_nat_bool(P,A4))
                 => pp(aa_set_nat_bool(P,insert_nat(B,A4))) ) ) )
         => pp(aa_set_nat_bool(P,A3)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_2505_finite__linorder__min__induct,axiom,
    ! [A3: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [B: int,A4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A4))
             => ( ! [X4: int] :
                    ( pp(aa_set_int_bool(member_int(X4),A4))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),X4)) )
               => ( pp(aa_set_int_bool(P,A4))
                 => pp(aa_set_int_bool(P,insert_int(B,A4))) ) ) )
         => pp(aa_set_int_bool(P,A3)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_2506_finite__subset__induct,axiom,
    ! [F2: set_set_nat,A3: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F2))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F2),A3))
       => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
         => ( ! [A: set_nat,F4: set_set_nat] :
                ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
               => ( pp(aa_set_set_nat_bool(member_set_nat(A),A3))
                 => ( ~ pp(aa_set_set_nat_bool(member_set_nat(A),F4))
                   => ( pp(aa_set_set_nat_bool(P,F4))
                     => pp(aa_set_set_nat_bool(P,insert_set_nat(A,F4))) ) ) ) )
           => pp(aa_set_set_nat_bool(P,F2)) ) ) ) ) ).

% finite_subset_induct
tff(fact_2507_finite__subset__induct,axiom,
    ! [F2: set_complex,A3: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F2),A3))
       => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
         => ( ! [A: complex,F4: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
               => ( pp(aa_set_complex_bool(member_complex(A),A3))
                 => ( ~ pp(aa_set_complex_bool(member_complex(A),F4))
                   => ( pp(aa_set_complex_bool(P,F4))
                     => pp(aa_set_complex_bool(P,insert_complex(A,F4))) ) ) ) )
           => pp(aa_set_complex_bool(P,F2)) ) ) ) ) ).

% finite_subset_induct
tff(fact_2508_finite__subset__induct,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F2))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(F2),A3))
       => ( pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat))
         => ( ! [A: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F4))
               => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A3))
                 => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),F4))
                   => ( pp(aa_set3135850010578026034t_bool(P,F4))
                     => pp(aa_set3135850010578026034t_bool(P,insert8211810215607154385at_nat(A,F4))) ) ) ) )
           => pp(aa_set3135850010578026034t_bool(P,F2)) ) ) ) ) ).

% finite_subset_induct
tff(fact_2509_finite__subset__induct,axiom,
    ! [F2: set_Extended_enat,A3: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(F2),A3))
       => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
         => ( ! [A: extended_enat,F4: set_Extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F4))
               => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A3))
                 => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),F4))
                   => ( pp(aa_set3347977610044113549t_bool(P,F4))
                     => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(A,F4))) ) ) ) )
           => pp(aa_set3347977610044113549t_bool(P,F2)) ) ) ) ) ).

% finite_subset_induct
tff(fact_2510_finite__subset__induct,axiom,
    ! [F2: set_real,A3: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(F2),A3))
       => ( pp(aa_set_real_bool(P,bot_bot_set_real))
         => ( ! [A: real,F4: set_real] :
                ( pp(aa_set_real_bool(finite_finite_real,F4))
               => ( pp(aa_set_real_bool(member_real(A),A3))
                 => ( ~ pp(aa_set_real_bool(member_real(A),F4))
                   => ( pp(aa_set_real_bool(P,F4))
                     => pp(aa_set_real_bool(P,insert_real(A,F4))) ) ) ) )
           => pp(aa_set_real_bool(P,F2)) ) ) ) ) ).

% finite_subset_induct
tff(fact_2511_finite__subset__induct,axiom,
    ! [F2: set_bool,A3: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,F2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(F2),A3))
       => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
         => ( ! [A: bool,F4: set_bool] :
                ( pp(aa_set_bool_bool(finite_finite_bool,F4))
               => ( pp(aa_set_bool_bool(member_bool(A),A3))
                 => ( ~ pp(aa_set_bool_bool(member_bool(A),F4))
                   => ( pp(aa_set_bool_bool(P,F4))
                     => pp(aa_set_bool_bool(P,insert_bool(A,F4))) ) ) ) )
           => pp(aa_set_bool_bool(P,F2)) ) ) ) ) ).

% finite_subset_induct
tff(fact_2512_finite__subset__induct,axiom,
    ! [F2: set_nat,A3: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,F2),A3))
       => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
         => ( ! [A: nat,F4: set_nat] :
                ( pp(aa_set_nat_bool(finite_finite_nat,F4))
               => ( pp(aa_set_nat_bool(member_nat(A),A3))
                 => ( ~ pp(aa_set_nat_bool(member_nat(A),F4))
                   => ( pp(aa_set_nat_bool(P,F4))
                     => pp(aa_set_nat_bool(P,insert_nat(A,F4))) ) ) ) )
           => pp(aa_set_nat_bool(P,F2)) ) ) ) ) ).

% finite_subset_induct
tff(fact_2513_finite__subset__induct,axiom,
    ! [F2: set_int,A3: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(F2),A3))
       => ( pp(aa_set_int_bool(P,bot_bot_set_int))
         => ( ! [A: int,F4: set_int] :
                ( pp(aa_set_int_bool(finite_finite_int,F4))
               => ( pp(aa_set_int_bool(member_int(A),A3))
                 => ( ~ pp(aa_set_int_bool(member_int(A),F4))
                   => ( pp(aa_set_int_bool(P,F4))
                     => pp(aa_set_int_bool(P,insert_int(A,F4))) ) ) ) )
           => pp(aa_set_int_bool(P,F2)) ) ) ) ) ).

% finite_subset_induct
tff(fact_2514_finite__subset__induct_H,axiom,
    ! [F2: set_set_nat,A3: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F2))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F2),A3))
       => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
         => ( ! [A: set_nat,F4: set_set_nat] :
                ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
               => ( pp(aa_set_set_nat_bool(member_set_nat(A),A3))
                 => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F4),A3))
                   => ( ~ pp(aa_set_set_nat_bool(member_set_nat(A),F4))
                     => ( pp(aa_set_set_nat_bool(P,F4))
                       => pp(aa_set_set_nat_bool(P,insert_set_nat(A,F4))) ) ) ) ) )
           => pp(aa_set_set_nat_bool(P,F2)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_2515_finite__subset__induct_H,axiom,
    ! [F2: set_complex,A3: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F2),A3))
       => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
         => ( ! [A: complex,F4: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
               => ( pp(aa_set_complex_bool(member_complex(A),A3))
                 => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F4),A3))
                   => ( ~ pp(aa_set_complex_bool(member_complex(A),F4))
                     => ( pp(aa_set_complex_bool(P,F4))
                       => pp(aa_set_complex_bool(P,insert_complex(A,F4))) ) ) ) ) )
           => pp(aa_set_complex_bool(P,F2)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_2516_finite__subset__induct_H,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F2))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(F2),A3))
       => ( pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat))
         => ( ! [A: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F4))
               => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A3))
                 => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(F4),A3))
                   => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),F4))
                     => ( pp(aa_set3135850010578026034t_bool(P,F4))
                       => pp(aa_set3135850010578026034t_bool(P,insert8211810215607154385at_nat(A,F4))) ) ) ) ) )
           => pp(aa_set3135850010578026034t_bool(P,F2)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_2517_finite__subset__induct_H,axiom,
    ! [F2: set_Extended_enat,A3: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(F2),A3))
       => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
         => ( ! [A: extended_enat,F4: set_Extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F4))
               => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A3))
                 => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(F4),A3))
                   => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),F4))
                     => ( pp(aa_set3347977610044113549t_bool(P,F4))
                       => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(A,F4))) ) ) ) ) )
           => pp(aa_set3347977610044113549t_bool(P,F2)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_2518_finite__subset__induct_H,axiom,
    ! [F2: set_real,A3: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(F2),A3))
       => ( pp(aa_set_real_bool(P,bot_bot_set_real))
         => ( ! [A: real,F4: set_real] :
                ( pp(aa_set_real_bool(finite_finite_real,F4))
               => ( pp(aa_set_real_bool(member_real(A),A3))
                 => ( pp(aa_set_real_bool(ord_less_eq_set_real(F4),A3))
                   => ( ~ pp(aa_set_real_bool(member_real(A),F4))
                     => ( pp(aa_set_real_bool(P,F4))
                       => pp(aa_set_real_bool(P,insert_real(A,F4))) ) ) ) ) )
           => pp(aa_set_real_bool(P,F2)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_2519_finite__subset__induct_H,axiom,
    ! [F2: set_bool,A3: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,F2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(F2),A3))
       => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
         => ( ! [A: bool,F4: set_bool] :
                ( pp(aa_set_bool_bool(finite_finite_bool,F4))
               => ( pp(aa_set_bool_bool(member_bool(A),A3))
                 => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(F4),A3))
                   => ( ~ pp(aa_set_bool_bool(member_bool(A),F4))
                     => ( pp(aa_set_bool_bool(P,F4))
                       => pp(aa_set_bool_bool(P,insert_bool(A,F4))) ) ) ) ) )
           => pp(aa_set_bool_bool(P,F2)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_2520_finite__subset__induct_H,axiom,
    ! [F2: set_nat,A3: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,F2),A3))
       => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
         => ( ! [A: nat,F4: set_nat] :
                ( pp(aa_set_nat_bool(finite_finite_nat,F4))
               => ( pp(aa_set_nat_bool(member_nat(A),A3))
                 => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,F4),A3))
                   => ( ~ pp(aa_set_nat_bool(member_nat(A),F4))
                     => ( pp(aa_set_nat_bool(P,F4))
                       => pp(aa_set_nat_bool(P,insert_nat(A,F4))) ) ) ) ) )
           => pp(aa_set_nat_bool(P,F2)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_2521_finite__subset__induct_H,axiom,
    ! [F2: set_int,A3: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(F2),A3))
       => ( pp(aa_set_int_bool(P,bot_bot_set_int))
         => ( ! [A: int,F4: set_int] :
                ( pp(aa_set_int_bool(finite_finite_int,F4))
               => ( pp(aa_set_int_bool(member_int(A),A3))
                 => ( pp(aa_set_int_bool(ord_less_eq_set_int(F4),A3))
                   => ( ~ pp(aa_set_int_bool(member_int(A),F4))
                     => ( pp(aa_set_int_bool(P,F4))
                       => pp(aa_set_int_bool(P,insert_int(A,F4))) ) ) ) ) )
           => pp(aa_set_int_bool(P,F2)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_2522_card__Suc__eq__finite,axiom,
    ! [A3: set_real,K: nat] :
      ( ( finite_card_real(A3) = aa_nat_nat(suc,K) )
    <=> ? [B6: real,B9: set_real] :
          ( ( A3 = insert_real(B6,B9) )
          & ~ pp(aa_set_real_bool(member_real(B6),B9))
          & ( finite_card_real(B9) = K )
          & pp(aa_set_real_bool(finite_finite_real,B9)) ) ) ).

% card_Suc_eq_finite
tff(fact_2523_card__Suc__eq__finite,axiom,
    ! [A3: set_bool,K: nat] :
      ( ( finite_card_bool(A3) = aa_nat_nat(suc,K) )
    <=> ? [B6: bool,B9: set_bool] :
          ( ( A3 = insert_bool(B6,B9) )
          & ~ pp(aa_set_bool_bool(member_bool(B6),B9))
          & ( finite_card_bool(B9) = K )
          & pp(aa_set_bool_bool(finite_finite_bool,B9)) ) ) ).

% card_Suc_eq_finite
tff(fact_2524_card__Suc__eq__finite,axiom,
    ! [A3: set_list_nat,K: nat] :
      ( ( finite_card_list_nat(A3) = aa_nat_nat(suc,K) )
    <=> ? [B6: list_nat,B9: set_list_nat] :
          ( ( A3 = insert_list_nat(B6,B9) )
          & ~ member_list_nat(B6,B9)
          & ( finite_card_list_nat(B9) = K )
          & finite8100373058378681591st_nat(B9) ) ) ).

% card_Suc_eq_finite
tff(fact_2525_card__Suc__eq__finite,axiom,
    ! [A3: set_set_nat,K: nat] :
      ( ( finite_card_set_nat(A3) = aa_nat_nat(suc,K) )
    <=> ? [B6: set_nat,B9: set_set_nat] :
          ( ( A3 = insert_set_nat(B6,B9) )
          & ~ pp(aa_set_set_nat_bool(member_set_nat(B6),B9))
          & ( finite_card_set_nat(B9) = K )
          & pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B9)) ) ) ).

% card_Suc_eq_finite
tff(fact_2526_card__Suc__eq__finite,axiom,
    ! [A3: set_nat,K: nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A3) = aa_nat_nat(suc,K) )
    <=> ? [B6: nat,B9: set_nat] :
          ( ( A3 = insert_nat(B6,B9) )
          & ~ pp(aa_set_nat_bool(member_nat(B6),B9))
          & ( aa_set_nat_nat(finite_card_nat,B9) = K )
          & pp(aa_set_nat_bool(finite_finite_nat,B9)) ) ) ).

% card_Suc_eq_finite
tff(fact_2527_card__Suc__eq__finite,axiom,
    ! [A3: set_int,K: nat] :
      ( ( finite_card_int(A3) = aa_nat_nat(suc,K) )
    <=> ? [B6: int,B9: set_int] :
          ( ( A3 = insert_int(B6,B9) )
          & ~ pp(aa_set_int_bool(member_int(B6),B9))
          & ( finite_card_int(B9) = K )
          & pp(aa_set_int_bool(finite_finite_int,B9)) ) ) ).

% card_Suc_eq_finite
tff(fact_2528_card__Suc__eq__finite,axiom,
    ! [A3: set_complex,K: nat] :
      ( ( finite_card_complex(A3) = aa_nat_nat(suc,K) )
    <=> ? [B6: complex,B9: set_complex] :
          ( ( A3 = insert_complex(B6,B9) )
          & ~ pp(aa_set_complex_bool(member_complex(B6),B9))
          & ( finite_card_complex(B9) = K )
          & pp(aa_set_complex_bool(finite3207457112153483333omplex,B9)) ) ) ).

% card_Suc_eq_finite
tff(fact_2529_card__Suc__eq__finite,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,K: nat] :
      ( ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3) = aa_nat_nat(suc,K) )
    <=> ? [B6: product_prod_nat_nat,B9: set_Pr1261947904930325089at_nat] :
          ( ( A3 = insert8211810215607154385at_nat(B6,B9) )
          & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B6),B9))
          & ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B9) = K )
          & pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B9)) ) ) ).

% card_Suc_eq_finite
tff(fact_2530_card__Suc__eq__finite,axiom,
    ! [A3: set_Extended_enat,K: nat] :
      ( ( finite121521170596916366d_enat(A3) = aa_nat_nat(suc,K) )
    <=> ? [B6: extended_enat,B9: set_Extended_enat] :
          ( ( A3 = insert_Extended_enat(B6,B9) )
          & ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(B6),B9))
          & ( finite121521170596916366d_enat(B9) = K )
          & pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B9)) ) ) ).

% card_Suc_eq_finite
tff(fact_2531_card__insert__if,axiom,
    ! [A3: set_real,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ( pp(aa_set_real_bool(member_real(X),A3))
         => ( finite_card_real(insert_real(X,A3)) = finite_card_real(A3) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A3))
         => ( finite_card_real(insert_real(X,A3)) = aa_nat_nat(suc,finite_card_real(A3)) ) ) ) ) ).

% card_insert_if
tff(fact_2532_card__insert__if,axiom,
    ! [A3: set_bool,X: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( ( pp(aa_set_bool_bool(member_bool(X),A3))
         => ( finite_card_bool(insert_bool(X,A3)) = finite_card_bool(A3) ) )
        & ( ~ pp(aa_set_bool_bool(member_bool(X),A3))
         => ( finite_card_bool(insert_bool(X,A3)) = aa_nat_nat(suc,finite_card_bool(A3)) ) ) ) ) ).

% card_insert_if
tff(fact_2533_card__insert__if,axiom,
    ! [A3: set_list_nat,X: list_nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( ( member_list_nat(X,A3)
         => ( finite_card_list_nat(insert_list_nat(X,A3)) = finite_card_list_nat(A3) ) )
        & ( ~ member_list_nat(X,A3)
         => ( finite_card_list_nat(insert_list_nat(X,A3)) = aa_nat_nat(suc,finite_card_list_nat(A3)) ) ) ) ) ).

% card_insert_if
tff(fact_2534_card__insert__if,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( ( pp(aa_set_set_nat_bool(member_set_nat(X),A3))
         => ( finite_card_set_nat(insert_set_nat(X,A3)) = finite_card_set_nat(A3) ) )
        & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A3))
         => ( finite_card_set_nat(insert_set_nat(X,A3)) = aa_nat_nat(suc,finite_card_set_nat(A3)) ) ) ) ) ).

% card_insert_if
tff(fact_2535_card__insert__if,axiom,
    ! [A3: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( pp(aa_set_nat_bool(member_nat(X),A3))
         => ( aa_set_nat_nat(finite_card_nat,insert_nat(X,A3)) = aa_set_nat_nat(finite_card_nat,A3) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A3))
         => ( aa_set_nat_nat(finite_card_nat,insert_nat(X,A3)) = aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,A3)) ) ) ) ) ).

% card_insert_if
tff(fact_2536_card__insert__if,axiom,
    ! [A3: set_int,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( pp(aa_set_int_bool(member_int(X),A3))
         => ( finite_card_int(insert_int(X,A3)) = finite_card_int(A3) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A3))
         => ( finite_card_int(insert_int(X,A3)) = aa_nat_nat(suc,finite_card_int(A3)) ) ) ) ) ).

% card_insert_if
tff(fact_2537_card__insert__if,axiom,
    ! [A3: set_complex,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A3))
         => ( finite_card_complex(insert_complex(X,A3)) = finite_card_complex(A3) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A3))
         => ( finite_card_complex(insert_complex(X,A3)) = aa_nat_nat(suc,finite_card_complex(A3)) ) ) ) ) ).

% card_insert_if
tff(fact_2538_card__insert__if,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
         => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,insert8211810215607154385at_nat(X,A3)) = aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3) ) )
        & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
         => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,insert8211810215607154385at_nat(X,A3)) = aa_nat_nat(suc,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)) ) ) ) ) ).

% card_insert_if
tff(fact_2539_card__insert__if,axiom,
    ! [A3: set_Extended_enat,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A3))
         => ( finite121521170596916366d_enat(insert_Extended_enat(X,A3)) = finite121521170596916366d_enat(A3) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A3))
         => ( finite121521170596916366d_enat(insert_Extended_enat(X,A3)) = aa_nat_nat(suc,finite121521170596916366d_enat(A3)) ) ) ) ) ).

% card_insert_if
tff(fact_2540_infinite__remove,axiom,
    ! [S: set_complex,A2: complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex)))) ) ).

% infinite_remove
tff(fact_2541_infinite__remove,axiom,
    ! [S: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S))
     => ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,S),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat)))) ) ).

% infinite_remove
tff(fact_2542_infinite__remove,axiom,
    ! [S: set_Extended_enat,A2: extended_enat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,minus_925952699566721837d_enat(S,insert_Extended_enat(A2,bot_bo7653980558646680370d_enat)))) ) ).

% infinite_remove
tff(fact_2543_infinite__remove,axiom,
    ! [S: set_real,A2: real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,S))
     => ~ pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(S,insert_real(A2,bot_bot_set_real)))) ) ).

% infinite_remove
tff(fact_2544_infinite__remove,axiom,
    ! [S: set_bool,A2: bool] :
      ( ~ pp(aa_set_bool_bool(finite_finite_bool,S))
     => ~ pp(aa_set_bool_bool(finite_finite_bool,minus_minus_set_bool(S,insert_bool(A2,bot_bot_set_bool)))) ) ).

% infinite_remove
tff(fact_2545_infinite__remove,axiom,
    ! [S: set_int,A2: int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S))
     => ~ pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),insert_int(A2,bot_bot_set_int)))) ) ).

% infinite_remove
tff(fact_2546_infinite__remove,axiom,
    ! [S: set_nat,A2: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
     => ~ pp(aa_set_nat_bool(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat(A2,bot_bot_set_nat)))) ) ).

% infinite_remove
tff(fact_2547_infinite__coinduct,axiom,
    ! [X6: fun_set_complex_bool,A3: set_complex] :
      ( pp(aa_set_complex_bool(X6,A3))
     => ( ! [A4: set_complex] :
            ( pp(aa_set_complex_bool(X6,A4))
           => ? [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),A4))
                & ( pp(aa_set_complex_bool(X6,minus_811609699411566653omplex(A4,insert_complex(X4,bot_bot_set_complex))))
                  | ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A4,insert_complex(X4,bot_bot_set_complex)))) ) ) )
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A3)) ) ) ).

% infinite_coinduct
tff(fact_2548_infinite__coinduct,axiom,
    ! [X6: fun_se8074476271188067385t_bool,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(X6,A3))
     => ( ! [A4: set_Pr1261947904930325089at_nat] :
            ( pp(aa_set3135850010578026034t_bool(X6,A4))
           => ? [X4: product_prod_nat_nat] :
                ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),A4))
                & ( pp(aa_set3135850010578026034t_bool(X6,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(X4,bot_bo2099793752762293965at_nat))))
                  | ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(X4,bot_bo2099793752762293965at_nat)))) ) ) )
       => ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3)) ) ) ).

% infinite_coinduct
tff(fact_2549_infinite__coinduct,axiom,
    ! [X6: fun_se2015444840911768724t_bool,A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(X6,A3))
     => ( ! [A4: set_Extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(X6,A4))
           => ? [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A4))
                & ( pp(aa_set3347977610044113549t_bool(X6,minus_925952699566721837d_enat(A4,insert_Extended_enat(X4,bot_bo7653980558646680370d_enat))))
                  | ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,minus_925952699566721837d_enat(A4,insert_Extended_enat(X4,bot_bo7653980558646680370d_enat)))) ) ) )
       => ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3)) ) ) ).

% infinite_coinduct
tff(fact_2550_infinite__coinduct,axiom,
    ! [X6: fun_set_real_bool,A3: set_real] :
      ( pp(aa_set_real_bool(X6,A3))
     => ( ! [A4: set_real] :
            ( pp(aa_set_real_bool(X6,A4))
           => ? [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),A4))
                & ( pp(aa_set_real_bool(X6,minus_minus_set_real(A4,insert_real(X4,bot_bot_set_real))))
                  | ~ pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A4,insert_real(X4,bot_bot_set_real)))) ) ) )
       => ~ pp(aa_set_real_bool(finite_finite_real,A3)) ) ) ).

% infinite_coinduct
tff(fact_2551_infinite__coinduct,axiom,
    ! [X6: fun_set_bool_bool,A3: set_bool] :
      ( pp(aa_set_bool_bool(X6,A3))
     => ( ! [A4: set_bool] :
            ( pp(aa_set_bool_bool(X6,A4))
           => ? [X4: bool] :
                ( pp(aa_set_bool_bool(member_bool(X4),A4))
                & ( pp(aa_set_bool_bool(X6,minus_minus_set_bool(A4,insert_bool(X4,bot_bot_set_bool))))
                  | ~ pp(aa_set_bool_bool(finite_finite_bool,minus_minus_set_bool(A4,insert_bool(X4,bot_bot_set_bool)))) ) ) )
       => ~ pp(aa_set_bool_bool(finite_finite_bool,A3)) ) ) ).

% infinite_coinduct
tff(fact_2552_infinite__coinduct,axiom,
    ! [X6: fun_set_int_bool,A3: set_int] :
      ( pp(aa_set_int_bool(X6,A3))
     => ( ! [A4: set_int] :
            ( pp(aa_set_int_bool(X6,A4))
           => ? [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),A4))
                & ( pp(aa_set_int_bool(X6,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(X4,bot_bot_set_int))))
                  | ~ pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(X4,bot_bot_set_int)))) ) ) )
       => ~ pp(aa_set_int_bool(finite_finite_int,A3)) ) ) ).

% infinite_coinduct
tff(fact_2553_infinite__coinduct,axiom,
    ! [X6: fun_set_nat_bool,A3: set_nat] :
      ( pp(aa_set_nat_bool(X6,A3))
     => ( ! [A4: set_nat] :
            ( pp(aa_set_nat_bool(X6,A4))
           => ? [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),A4))
                & ( pp(aa_set_nat_bool(X6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X4,bot_bot_set_nat))))
                  | ~ pp(aa_set_nat_bool(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X4,bot_bot_set_nat)))) ) ) )
       => ~ pp(aa_set_nat_bool(finite_finite_nat,A3)) ) ) ).

% infinite_coinduct
tff(fact_2554_finite__empty__induct,axiom,
    ! [A3: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( pp(aa_set_set_nat_bool(P,A3))
       => ( ! [A: set_nat,A4: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
             => ( pp(aa_set_set_nat_bool(member_set_nat(A),A4))
               => ( pp(aa_set_set_nat_bool(P,A4))
                 => pp(aa_set_set_nat_bool(P,minus_2163939370556025621et_nat(A4,insert_set_nat(A,bot_bot_set_set_nat)))) ) ) )
         => pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_2555_finite__empty__induct,axiom,
    ! [A3: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(P,A3))
       => ( ! [A: complex,A4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
             => ( pp(aa_set_complex_bool(member_complex(A),A4))
               => ( pp(aa_set_complex_bool(P,A4))
                 => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex)))) ) ) )
         => pp(aa_set_complex_bool(P,bot_bot_set_complex)) ) ) ) ).

% finite_empty_induct
tff(fact_2556_finite__empty__induct,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( pp(aa_set3135850010578026034t_bool(P,A3))
       => ( ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
             => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A4))
               => ( pp(aa_set3135850010578026034t_bool(P,A4))
                 => pp(aa_set3135850010578026034t_bool(P,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)))) ) ) )
         => pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_2557_finite__empty__induct,axiom,
    ! [A3: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(P,A3))
       => ( ! [A: extended_enat,A4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A4))
             => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A4))
               => ( pp(aa_set3347977610044113549t_bool(P,A4))
                 => pp(aa_set3347977610044113549t_bool(P,minus_925952699566721837d_enat(A4,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat)) ) ) ) ).

% finite_empty_induct
tff(fact_2558_finite__empty__induct,axiom,
    ! [A3: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(P,A3))
       => ( ! [A: real,A4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A4))
             => ( pp(aa_set_real_bool(member_real(A),A4))
               => ( pp(aa_set_real_bool(P,A4))
                 => pp(aa_set_real_bool(P,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real)))) ) ) )
         => pp(aa_set_real_bool(P,bot_bot_set_real)) ) ) ) ).

% finite_empty_induct
tff(fact_2559_finite__empty__induct,axiom,
    ! [A3: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( pp(aa_set_bool_bool(P,A3))
       => ( ! [A: bool,A4: set_bool] :
              ( pp(aa_set_bool_bool(finite_finite_bool,A4))
             => ( pp(aa_set_bool_bool(member_bool(A),A4))
               => ( pp(aa_set_bool_bool(P,A4))
                 => pp(aa_set_bool_bool(P,minus_minus_set_bool(A4,insert_bool(A,bot_bot_set_bool)))) ) ) )
         => pp(aa_set_bool_bool(P,bot_bot_set_bool)) ) ) ) ).

% finite_empty_induct
tff(fact_2560_finite__empty__induct,axiom,
    ! [A3: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(P,A3))
       => ( ! [A: int,A4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A4))
             => ( pp(aa_set_int_bool(member_int(A),A4))
               => ( pp(aa_set_int_bool(P,A4))
                 => pp(aa_set_int_bool(P,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(A,bot_bot_set_int)))) ) ) )
         => pp(aa_set_int_bool(P,bot_bot_set_int)) ) ) ) ).

% finite_empty_induct
tff(fact_2561_finite__empty__induct,axiom,
    ! [A3: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(P,A3))
       => ( ! [A: nat,A4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A4))
             => ( pp(aa_set_nat_bool(member_nat(A),A4))
               => ( pp(aa_set_nat_bool(P,A4))
                 => pp(aa_set_nat_bool(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,bot_bot_set_nat)))) ) ) )
         => pp(aa_set_nat_bool(P,bot_bot_set_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_2562_subset__insert__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),insert8211810215607154385at_nat(X,B2)))
    <=> ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
         => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B2)) )
        & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
         => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),B2)) ) ) ) ).

% subset_insert_iff
tff(fact_2563_subset__insert__iff,axiom,
    ! [A3: set_set_nat,X: set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),insert_set_nat(X,B2)))
    <=> ( ( pp(aa_set_set_nat_bool(member_set_nat(X),A3))
         => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(A3,insert_set_nat(X,bot_bot_set_set_nat))),B2)) )
        & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A3))
         => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),B2)) ) ) ) ).

% subset_insert_iff
tff(fact_2564_subset__insert__iff,axiom,
    ! [A3: set_real,X: real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),insert_real(X,B2)))
    <=> ( ( pp(aa_set_real_bool(member_real(X),A3))
         => pp(aa_set_real_bool(ord_less_eq_set_real(minus_minus_set_real(A3,insert_real(X,bot_bot_set_real))),B2)) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A3))
         => pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2)) ) ) ) ).

% subset_insert_iff
tff(fact_2565_subset__insert__iff,axiom,
    ! [A3: set_bool,X: bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),insert_bool(X,B2)))
    <=> ( ( pp(aa_set_bool_bool(member_bool(X),A3))
         => pp(aa_set_bool_bool(ord_less_eq_set_bool(minus_minus_set_bool(A3,insert_bool(X,bot_bot_set_bool))),B2)) )
        & ( ~ pp(aa_set_bool_bool(member_bool(X),A3))
         => pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),B2)) ) ) ) ).

% subset_insert_iff
tff(fact_2566_subset__insert__iff,axiom,
    ! [A3: set_nat,X: nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),insert_nat(X,B2)))
    <=> ( ( pp(aa_set_nat_bool(member_nat(X),A3))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(X,bot_bot_set_nat))),B2)) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A3))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2)) ) ) ) ).

% subset_insert_iff
tff(fact_2567_subset__insert__iff,axiom,
    ! [A3: set_int,X: int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),insert_int(X,B2)))
    <=> ( ( pp(aa_set_int_bool(member_int(X),A3))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(X,bot_bot_set_int))),B2)) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A3))
         => pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2)) ) ) ) ).

% subset_insert_iff
tff(fact_2568_Diff__single__insert,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B2))
     => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),insert8211810215607154385at_nat(X,B2))) ) ).

% Diff_single_insert
tff(fact_2569_Diff__single__insert,axiom,
    ! [A3: set_real,X: real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(minus_minus_set_real(A3,insert_real(X,bot_bot_set_real))),B2))
     => pp(aa_set_real_bool(ord_less_eq_set_real(A3),insert_real(X,B2))) ) ).

% Diff_single_insert
tff(fact_2570_Diff__single__insert,axiom,
    ! [A3: set_bool,X: bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(minus_minus_set_bool(A3,insert_bool(X,bot_bot_set_bool))),B2))
     => pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),insert_bool(X,B2))) ) ).

% Diff_single_insert
tff(fact_2571_Diff__single__insert,axiom,
    ! [A3: set_nat,X: nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(X,bot_bot_set_nat))),B2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),insert_nat(X,B2))) ) ).

% Diff_single_insert
tff(fact_2572_Diff__single__insert,axiom,
    ! [A3: set_int,X: int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(X,bot_bot_set_int))),B2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A3),insert_int(X,B2))) ) ).

% Diff_single_insert
tff(fact_2573_card__1__singletonE,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3) = one_one_nat )
     => ~ ! [X5: product_prod_nat_nat] : A3 != insert8211810215607154385at_nat(X5,bot_bo2099793752762293965at_nat) ) ).

% card_1_singletonE
tff(fact_2574_card__1__singletonE,axiom,
    ! [A3: set_complex] :
      ( ( finite_card_complex(A3) = one_one_nat )
     => ~ ! [X5: complex] : A3 != insert_complex(X5,bot_bot_set_complex) ) ).

% card_1_singletonE
tff(fact_2575_card__1__singletonE,axiom,
    ! [A3: set_list_nat] :
      ( ( finite_card_list_nat(A3) = one_one_nat )
     => ~ ! [X5: list_nat] : A3 != insert_list_nat(X5,bot_bot_set_list_nat) ) ).

% card_1_singletonE
tff(fact_2576_card__1__singletonE,axiom,
    ! [A3: set_set_nat] :
      ( ( finite_card_set_nat(A3) = one_one_nat )
     => ~ ! [X5: set_nat] : A3 != insert_set_nat(X5,bot_bot_set_set_nat) ) ).

% card_1_singletonE
tff(fact_2577_card__1__singletonE,axiom,
    ! [A3: set_real] :
      ( ( finite_card_real(A3) = one_one_nat )
     => ~ ! [X5: real] : A3 != insert_real(X5,bot_bot_set_real) ) ).

% card_1_singletonE
tff(fact_2578_card__1__singletonE,axiom,
    ! [A3: set_bool] :
      ( ( finite_card_bool(A3) = one_one_nat )
     => ~ ! [X5: bool] : A3 != insert_bool(X5,bot_bot_set_bool) ) ).

% card_1_singletonE
tff(fact_2579_card__1__singletonE,axiom,
    ! [A3: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A3) = one_one_nat )
     => ~ ! [X5: nat] : A3 != insert_nat(X5,bot_bot_set_nat) ) ).

% card_1_singletonE
tff(fact_2580_card__1__singletonE,axiom,
    ! [A3: set_int] :
      ( ( finite_card_int(A3) = one_one_nat )
     => ~ ! [X5: int] : A3 != insert_int(X5,bot_bot_set_int) ) ).

% card_1_singletonE
tff(fact_2581_Compl__insert,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(insert8211810215607154385at_nat(X,A3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,uminus6524753893492686040at_nat(A3)),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) ).

% Compl_insert
tff(fact_2582_Compl__insert,axiom,
    ! [X: real,A3: set_real] : uminus612125837232591019t_real(insert_real(X,A3)) = minus_minus_set_real(uminus612125837232591019t_real(A3),insert_real(X,bot_bot_set_real)) ).

% Compl_insert
tff(fact_2583_Compl__insert,axiom,
    ! [X: bool,A3: set_bool] : uminus5166143097222041326t_bool(insert_bool(X,A3)) = minus_minus_set_bool(uminus5166143097222041326t_bool(A3),insert_bool(X,bot_bot_set_bool)) ).

% Compl_insert
tff(fact_2584_Compl__insert,axiom,
    ! [X: int,A3: set_int] : uminus1532241313380277803et_int(insert_int(X,A3)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,uminus1532241313380277803et_int(A3)),insert_int(X,bot_bot_set_int)) ).

% Compl_insert
tff(fact_2585_Compl__insert,axiom,
    ! [X: nat,A3: set_nat] : uminus5710092332889474511et_nat(insert_nat(X,A3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,uminus5710092332889474511et_nat(A3)),insert_nat(X,bot_bot_set_nat)) ).

% Compl_insert
tff(fact_2586_card__Suc__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,K: nat] :
      ( ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3) = aa_nat_nat(suc,K) )
    <=> ? [B6: product_prod_nat_nat,B9: set_Pr1261947904930325089at_nat] :
          ( ( A3 = insert8211810215607154385at_nat(B6,B9) )
          & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B6),B9))
          & ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B9) = K )
          & ( ( K = zero_zero_nat )
           => ( B9 = bot_bo2099793752762293965at_nat ) ) ) ) ).

% card_Suc_eq
tff(fact_2587_card__Suc__eq,axiom,
    ! [A3: set_complex,K: nat] :
      ( ( finite_card_complex(A3) = aa_nat_nat(suc,K) )
    <=> ? [B6: complex,B9: set_complex] :
          ( ( A3 = insert_complex(B6,B9) )
          & ~ pp(aa_set_complex_bool(member_complex(B6),B9))
          & ( finite_card_complex(B9) = K )
          & ( ( K = zero_zero_nat )
           => ( B9 = bot_bot_set_complex ) ) ) ) ).

% card_Suc_eq
tff(fact_2588_card__Suc__eq,axiom,
    ! [A3: set_list_nat,K: nat] :
      ( ( finite_card_list_nat(A3) = aa_nat_nat(suc,K) )
    <=> ? [B6: list_nat,B9: set_list_nat] :
          ( ( A3 = insert_list_nat(B6,B9) )
          & ~ member_list_nat(B6,B9)
          & ( finite_card_list_nat(B9) = K )
          & ( ( K = zero_zero_nat )
           => ( B9 = bot_bot_set_list_nat ) ) ) ) ).

% card_Suc_eq
tff(fact_2589_card__Suc__eq,axiom,
    ! [A3: set_set_nat,K: nat] :
      ( ( finite_card_set_nat(A3) = aa_nat_nat(suc,K) )
    <=> ? [B6: set_nat,B9: set_set_nat] :
          ( ( A3 = insert_set_nat(B6,B9) )
          & ~ pp(aa_set_set_nat_bool(member_set_nat(B6),B9))
          & ( finite_card_set_nat(B9) = K )
          & ( ( K = zero_zero_nat )
           => ( B9 = bot_bot_set_set_nat ) ) ) ) ).

% card_Suc_eq
tff(fact_2590_card__Suc__eq,axiom,
    ! [A3: set_real,K: nat] :
      ( ( finite_card_real(A3) = aa_nat_nat(suc,K) )
    <=> ? [B6: real,B9: set_real] :
          ( ( A3 = insert_real(B6,B9) )
          & ~ pp(aa_set_real_bool(member_real(B6),B9))
          & ( finite_card_real(B9) = K )
          & ( ( K = zero_zero_nat )
           => ( B9 = bot_bot_set_real ) ) ) ) ).

% card_Suc_eq
tff(fact_2591_card__Suc__eq,axiom,
    ! [A3: set_bool,K: nat] :
      ( ( finite_card_bool(A3) = aa_nat_nat(suc,K) )
    <=> ? [B6: bool,B9: set_bool] :
          ( ( A3 = insert_bool(B6,B9) )
          & ~ pp(aa_set_bool_bool(member_bool(B6),B9))
          & ( finite_card_bool(B9) = K )
          & ( ( K = zero_zero_nat )
           => ( B9 = bot_bot_set_bool ) ) ) ) ).

% card_Suc_eq
tff(fact_2592_card__Suc__eq,axiom,
    ! [A3: set_nat,K: nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A3) = aa_nat_nat(suc,K) )
    <=> ? [B6: nat,B9: set_nat] :
          ( ( A3 = insert_nat(B6,B9) )
          & ~ pp(aa_set_nat_bool(member_nat(B6),B9))
          & ( aa_set_nat_nat(finite_card_nat,B9) = K )
          & ( ( K = zero_zero_nat )
           => ( B9 = bot_bot_set_nat ) ) ) ) ).

% card_Suc_eq
tff(fact_2593_card__Suc__eq,axiom,
    ! [A3: set_int,K: nat] :
      ( ( finite_card_int(A3) = aa_nat_nat(suc,K) )
    <=> ? [B6: int,B9: set_int] :
          ( ( A3 = insert_int(B6,B9) )
          & ~ pp(aa_set_int_bool(member_int(B6),B9))
          & ( finite_card_int(B9) = K )
          & ( ( K = zero_zero_nat )
           => ( B9 = bot_bot_set_int ) ) ) ) ).

% card_Suc_eq
tff(fact_2594_card__eq__SucD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,K: nat] :
      ( ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3) = aa_nat_nat(suc,K) )
     => ? [B: product_prod_nat_nat,B7: set_Pr1261947904930325089at_nat] :
          ( ( A3 = insert8211810215607154385at_nat(B,B7) )
          & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B),B7))
          & ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bo2099793752762293965at_nat ) ) ) ) ).

% card_eq_SucD
tff(fact_2595_card__eq__SucD,axiom,
    ! [A3: set_complex,K: nat] :
      ( ( finite_card_complex(A3) = aa_nat_nat(suc,K) )
     => ? [B: complex,B7: set_complex] :
          ( ( A3 = insert_complex(B,B7) )
          & ~ pp(aa_set_complex_bool(member_complex(B),B7))
          & ( finite_card_complex(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_complex ) ) ) ) ).

% card_eq_SucD
tff(fact_2596_card__eq__SucD,axiom,
    ! [A3: set_list_nat,K: nat] :
      ( ( finite_card_list_nat(A3) = aa_nat_nat(suc,K) )
     => ? [B: list_nat,B7: set_list_nat] :
          ( ( A3 = insert_list_nat(B,B7) )
          & ~ member_list_nat(B,B7)
          & ( finite_card_list_nat(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_list_nat ) ) ) ) ).

% card_eq_SucD
tff(fact_2597_card__eq__SucD,axiom,
    ! [A3: set_set_nat,K: nat] :
      ( ( finite_card_set_nat(A3) = aa_nat_nat(suc,K) )
     => ? [B: set_nat,B7: set_set_nat] :
          ( ( A3 = insert_set_nat(B,B7) )
          & ~ pp(aa_set_set_nat_bool(member_set_nat(B),B7))
          & ( finite_card_set_nat(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_set_nat ) ) ) ) ).

% card_eq_SucD
tff(fact_2598_card__eq__SucD,axiom,
    ! [A3: set_real,K: nat] :
      ( ( finite_card_real(A3) = aa_nat_nat(suc,K) )
     => ? [B: real,B7: set_real] :
          ( ( A3 = insert_real(B,B7) )
          & ~ pp(aa_set_real_bool(member_real(B),B7))
          & ( finite_card_real(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_real ) ) ) ) ).

% card_eq_SucD
tff(fact_2599_card__eq__SucD,axiom,
    ! [A3: set_bool,K: nat] :
      ( ( finite_card_bool(A3) = aa_nat_nat(suc,K) )
     => ? [B: bool,B7: set_bool] :
          ( ( A3 = insert_bool(B,B7) )
          & ~ pp(aa_set_bool_bool(member_bool(B),B7))
          & ( finite_card_bool(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_bool ) ) ) ) ).

% card_eq_SucD
tff(fact_2600_card__eq__SucD,axiom,
    ! [A3: set_nat,K: nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A3) = aa_nat_nat(suc,K) )
     => ? [B: nat,B7: set_nat] :
          ( ( A3 = insert_nat(B,B7) )
          & ~ pp(aa_set_nat_bool(member_nat(B),B7))
          & ( aa_set_nat_nat(finite_card_nat,B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_nat ) ) ) ) ).

% card_eq_SucD
tff(fact_2601_card__eq__SucD,axiom,
    ! [A3: set_int,K: nat] :
      ( ( finite_card_int(A3) = aa_nat_nat(suc,K) )
     => ? [B: int,B7: set_int] :
          ( ( A3 = insert_int(B,B7) )
          & ~ pp(aa_set_int_bool(member_int(B),B7))
          & ( finite_card_int(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_int ) ) ) ) ).

% card_eq_SucD
tff(fact_2602_card__1__singleton__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X3: product_prod_nat_nat] : A3 = insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat) ) ).

% card_1_singleton_iff
tff(fact_2603_card__1__singleton__iff,axiom,
    ! [A3: set_complex] :
      ( ( finite_card_complex(A3) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X3: complex] : A3 = insert_complex(X3,bot_bot_set_complex) ) ).

% card_1_singleton_iff
tff(fact_2604_card__1__singleton__iff,axiom,
    ! [A3: set_list_nat] :
      ( ( finite_card_list_nat(A3) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X3: list_nat] : A3 = insert_list_nat(X3,bot_bot_set_list_nat) ) ).

% card_1_singleton_iff
tff(fact_2605_card__1__singleton__iff,axiom,
    ! [A3: set_set_nat] :
      ( ( finite_card_set_nat(A3) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X3: set_nat] : A3 = insert_set_nat(X3,bot_bot_set_set_nat) ) ).

% card_1_singleton_iff
tff(fact_2606_card__1__singleton__iff,axiom,
    ! [A3: set_real] :
      ( ( finite_card_real(A3) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X3: real] : A3 = insert_real(X3,bot_bot_set_real) ) ).

% card_1_singleton_iff
tff(fact_2607_card__1__singleton__iff,axiom,
    ! [A3: set_bool] :
      ( ( finite_card_bool(A3) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X3: bool] : A3 = insert_bool(X3,bot_bot_set_bool) ) ).

% card_1_singleton_iff
tff(fact_2608_card__1__singleton__iff,axiom,
    ! [A3: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A3) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X3: nat] : A3 = insert_nat(X3,bot_bot_set_nat) ) ).

% card_1_singleton_iff
tff(fact_2609_card__1__singleton__iff,axiom,
    ! [A3: set_int] :
      ( ( finite_card_int(A3) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X3: int] : A3 = insert_int(X3,bot_bot_set_int) ) ).

% card_1_singleton_iff
tff(fact_2610_remove__induct,axiom,
    ! [P: fun_set_set_nat_bool,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
     => ( ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B2))
         => pp(aa_set_set_nat_bool(P,B2)) )
       => ( ! [A4: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
             => ( ( A4 != bot_bot_set_set_nat )
               => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A4),B2))
                 => ( ! [X4: set_nat] :
                        ( pp(aa_set_set_nat_bool(member_set_nat(X4),A4))
                       => pp(aa_set_set_nat_bool(P,minus_2163939370556025621et_nat(A4,insert_set_nat(X4,bot_bot_set_set_nat)))) )
                   => pp(aa_set_set_nat_bool(P,A4)) ) ) ) )
         => pp(aa_set_set_nat_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_2611_remove__induct,axiom,
    ! [P: fun_set_complex_bool,B2: set_complex] :
      ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
     => ( ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
         => pp(aa_set_complex_bool(P,B2)) )
       => ( ! [A4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
             => ( ( A4 != bot_bot_set_complex )
               => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B2))
                 => ( ! [X4: complex] :
                        ( pp(aa_set_complex_bool(member_complex(X4),A4))
                       => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A4,insert_complex(X4,bot_bot_set_complex)))) )
                   => pp(aa_set_complex_bool(P,A4)) ) ) ) )
         => pp(aa_set_complex_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_2612_remove__induct,axiom,
    ! [P: fun_se8074476271188067385t_bool,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat))
     => ( ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
         => pp(aa_set3135850010578026034t_bool(P,B2)) )
       => ( ! [A4: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
             => ( ( A4 != bot_bo2099793752762293965at_nat )
               => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A4),B2))
                 => ( ! [X4: product_prod_nat_nat] :
                        ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),A4))
                       => pp(aa_set3135850010578026034t_bool(P,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(X4,bot_bo2099793752762293965at_nat)))) )
                   => pp(aa_set3135850010578026034t_bool(P,A4)) ) ) ) )
         => pp(aa_set3135850010578026034t_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_2613_remove__induct,axiom,
    ! [P: fun_se2015444840911768724t_bool,B2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
     => ( ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
         => pp(aa_set3347977610044113549t_bool(P,B2)) )
       => ( ! [A4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A4))
             => ( ( A4 != bot_bo7653980558646680370d_enat )
               => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A4),B2))
                 => ( ! [X4: extended_enat] :
                        ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A4))
                       => pp(aa_set3347977610044113549t_bool(P,minus_925952699566721837d_enat(A4,insert_Extended_enat(X4,bot_bo7653980558646680370d_enat)))) )
                   => pp(aa_set3347977610044113549t_bool(P,A4)) ) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_2614_remove__induct,axiom,
    ! [P: fun_set_real_bool,B2: set_real] :
      ( pp(aa_set_real_bool(P,bot_bot_set_real))
     => ( ( ~ pp(aa_set_real_bool(finite_finite_real,B2))
         => pp(aa_set_real_bool(P,B2)) )
       => ( ! [A4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A4))
             => ( ( A4 != bot_bot_set_real )
               => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B2))
                 => ( ! [X4: real] :
                        ( pp(aa_set_real_bool(member_real(X4),A4))
                       => pp(aa_set_real_bool(P,minus_minus_set_real(A4,insert_real(X4,bot_bot_set_real)))) )
                   => pp(aa_set_real_bool(P,A4)) ) ) ) )
         => pp(aa_set_real_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_2615_remove__induct,axiom,
    ! [P: fun_set_bool_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
     => ( ( ~ pp(aa_set_bool_bool(finite_finite_bool,B2))
         => pp(aa_set_bool_bool(P,B2)) )
       => ( ! [A4: set_bool] :
              ( pp(aa_set_bool_bool(finite_finite_bool,A4))
             => ( ( A4 != bot_bot_set_bool )
               => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A4),B2))
                 => ( ! [X4: bool] :
                        ( pp(aa_set_bool_bool(member_bool(X4),A4))
                       => pp(aa_set_bool_bool(P,minus_minus_set_bool(A4,insert_bool(X4,bot_bot_set_bool)))) )
                   => pp(aa_set_bool_bool(P,A4)) ) ) ) )
         => pp(aa_set_bool_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_2616_remove__induct,axiom,
    ! [P: fun_set_nat_bool,B2: set_nat] :
      ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
     => ( ( ~ pp(aa_set_nat_bool(finite_finite_nat,B2))
         => pp(aa_set_nat_bool(P,B2)) )
       => ( ! [A4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A4))
             => ( ( A4 != bot_bot_set_nat )
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B2))
                 => ( ! [X4: nat] :
                        ( pp(aa_set_nat_bool(member_nat(X4),A4))
                       => pp(aa_set_nat_bool(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X4,bot_bot_set_nat)))) )
                   => pp(aa_set_nat_bool(P,A4)) ) ) ) )
         => pp(aa_set_nat_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_2617_remove__induct,axiom,
    ! [P: fun_set_int_bool,B2: set_int] :
      ( pp(aa_set_int_bool(P,bot_bot_set_int))
     => ( ( ~ pp(aa_set_int_bool(finite_finite_int,B2))
         => pp(aa_set_int_bool(P,B2)) )
       => ( ! [A4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A4))
             => ( ( A4 != bot_bot_set_int )
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B2))
                 => ( ! [X4: int] :
                        ( pp(aa_set_int_bool(member_int(X4),A4))
                       => pp(aa_set_int_bool(P,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(X4,bot_bot_set_int)))) )
                   => pp(aa_set_int_bool(P,A4)) ) ) ) )
         => pp(aa_set_int_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_2618_finite__remove__induct,axiom,
    ! [B2: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B2))
     => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
       => ( ! [A4: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
             => ( ( A4 != bot_bot_set_set_nat )
               => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A4),B2))
                 => ( ! [X4: set_nat] :
                        ( pp(aa_set_set_nat_bool(member_set_nat(X4),A4))
                       => pp(aa_set_set_nat_bool(P,minus_2163939370556025621et_nat(A4,insert_set_nat(X4,bot_bot_set_set_nat)))) )
                   => pp(aa_set_set_nat_bool(P,A4)) ) ) ) )
         => pp(aa_set_set_nat_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_2619_finite__remove__induct,axiom,
    ! [B2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [A4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
             => ( ( A4 != bot_bot_set_complex )
               => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B2))
                 => ( ! [X4: complex] :
                        ( pp(aa_set_complex_bool(member_complex(X4),A4))
                       => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A4,insert_complex(X4,bot_bot_set_complex)))) )
                   => pp(aa_set_complex_bool(P,A4)) ) ) ) )
         => pp(aa_set_complex_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_2620_finite__remove__induct,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
     => ( pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat))
       => ( ! [A4: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
             => ( ( A4 != bot_bo2099793752762293965at_nat )
               => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A4),B2))
                 => ( ! [X4: product_prod_nat_nat] :
                        ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),A4))
                       => pp(aa_set3135850010578026034t_bool(P,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(X4,bot_bo2099793752762293965at_nat)))) )
                   => pp(aa_set3135850010578026034t_bool(P,A4)) ) ) ) )
         => pp(aa_set3135850010578026034t_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_2621_finite__remove__induct,axiom,
    ! [B2: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [A4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A4))
             => ( ( A4 != bot_bo7653980558646680370d_enat )
               => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A4),B2))
                 => ( ! [X4: extended_enat] :
                        ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A4))
                       => pp(aa_set3347977610044113549t_bool(P,minus_925952699566721837d_enat(A4,insert_Extended_enat(X4,bot_bo7653980558646680370d_enat)))) )
                   => pp(aa_set3347977610044113549t_bool(P,A4)) ) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_2622_finite__remove__induct,axiom,
    ! [B2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [A4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A4))
             => ( ( A4 != bot_bot_set_real )
               => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B2))
                 => ( ! [X4: real] :
                        ( pp(aa_set_real_bool(member_real(X4),A4))
                       => pp(aa_set_real_bool(P,minus_minus_set_real(A4,insert_real(X4,bot_bot_set_real)))) )
                   => pp(aa_set_real_bool(P,A4)) ) ) ) )
         => pp(aa_set_real_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_2623_finite__remove__induct,axiom,
    ! [B2: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,B2))
     => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
       => ( ! [A4: set_bool] :
              ( pp(aa_set_bool_bool(finite_finite_bool,A4))
             => ( ( A4 != bot_bot_set_bool )
               => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A4),B2))
                 => ( ! [X4: bool] :
                        ( pp(aa_set_bool_bool(member_bool(X4),A4))
                       => pp(aa_set_bool_bool(P,minus_minus_set_bool(A4,insert_bool(X4,bot_bot_set_bool)))) )
                   => pp(aa_set_bool_bool(P,A4)) ) ) ) )
         => pp(aa_set_bool_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_2624_finite__remove__induct,axiom,
    ! [B2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [A4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A4))
             => ( ( A4 != bot_bot_set_nat )
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B2))
                 => ( ! [X4: nat] :
                        ( pp(aa_set_nat_bool(member_nat(X4),A4))
                       => pp(aa_set_nat_bool(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X4,bot_bot_set_nat)))) )
                   => pp(aa_set_nat_bool(P,A4)) ) ) ) )
         => pp(aa_set_nat_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_2625_finite__remove__induct,axiom,
    ! [B2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [A4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A4))
             => ( ( A4 != bot_bot_set_int )
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B2))
                 => ( ! [X4: int] :
                        ( pp(aa_set_int_bool(member_int(X4),A4))
                       => pp(aa_set_int_bool(P,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(X4,bot_bot_set_int)))) )
                   => pp(aa_set_int_bool(P,A4)) ) ) ) )
         => pp(aa_set_int_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_2626_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),finite_card_real(A3)))
    <=> ? [A6: real,B9: set_real] :
          ( ( A3 = insert_real(A6,B9) )
          & ~ pp(aa_set_real_bool(member_real(A6),B9))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_real(B9)))
          & pp(aa_set_real_bool(finite_finite_real,B9)) ) ) ).

% card_le_Suc_iff
tff(fact_2627_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),finite_card_bool(A3)))
    <=> ? [A6: bool,B9: set_bool] :
          ( ( A3 = insert_bool(A6,B9) )
          & ~ pp(aa_set_bool_bool(member_bool(A6),B9))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_bool(B9)))
          & pp(aa_set_bool_bool(finite_finite_bool,B9)) ) ) ).

% card_le_Suc_iff
tff(fact_2628_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),finite_card_list_nat(A3)))
    <=> ? [A6: list_nat,B9: set_list_nat] :
          ( ( A3 = insert_list_nat(A6,B9) )
          & ~ member_list_nat(A6,B9)
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_list_nat(B9)))
          & finite8100373058378681591st_nat(B9) ) ) ).

% card_le_Suc_iff
tff(fact_2629_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),finite_card_set_nat(A3)))
    <=> ? [A6: set_nat,B9: set_set_nat] :
          ( ( A3 = insert_set_nat(A6,B9) )
          & ~ pp(aa_set_set_nat_bool(member_set_nat(A6),B9))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_set_nat(B9)))
          & pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B9)) ) ) ).

% card_le_Suc_iff
tff(fact_2630_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_set_nat_nat(finite_card_nat,A3)))
    <=> ? [A6: nat,B9: set_nat] :
          ( ( A3 = insert_nat(A6,B9) )
          & ~ pp(aa_set_nat_bool(member_nat(A6),B9))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_set_nat_nat(finite_card_nat,B9)))
          & pp(aa_set_nat_bool(finite_finite_nat,B9)) ) ) ).

% card_le_Suc_iff
tff(fact_2631_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),finite_card_int(A3)))
    <=> ? [A6: int,B9: set_int] :
          ( ( A3 = insert_int(A6,B9) )
          & ~ pp(aa_set_int_bool(member_int(A6),B9))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_int(B9)))
          & pp(aa_set_int_bool(finite_finite_int,B9)) ) ) ).

% card_le_Suc_iff
tff(fact_2632_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),finite_card_complex(A3)))
    <=> ? [A6: complex,B9: set_complex] :
          ( ( A3 = insert_complex(A6,B9) )
          & ~ pp(aa_set_complex_bool(member_complex(A6),B9))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_complex(B9)))
          & pp(aa_set_complex_bool(finite3207457112153483333omplex,B9)) ) ) ).

% card_le_Suc_iff
tff(fact_2633_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)))
    <=> ? [A6: product_prod_nat_nat,B9: set_Pr1261947904930325089at_nat] :
          ( ( A3 = insert8211810215607154385at_nat(A6,B9) )
          & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A6),B9))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B9)))
          & pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B9)) ) ) ).

% card_le_Suc_iff
tff(fact_2634_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_Extended_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),finite121521170596916366d_enat(A3)))
    <=> ? [A6: extended_enat,B9: set_Extended_enat] :
          ( ( A3 = insert_Extended_enat(A6,B9) )
          & ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A6),B9))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite121521170596916366d_enat(B9)))
          & pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B9)) ) ) ).

% card_le_Suc_iff
tff(fact_2635_card__Diff1__le,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)))),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3))) ).

% card_Diff1_le
tff(fact_2636_card__Diff1__le,axiom,
    ! [A3: set_complex,X: complex] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex)))),finite_card_complex(A3))) ).

% card_Diff1_le
tff(fact_2637_card__Diff1__le,axiom,
    ! [A3: set_list_nat,X: list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A3,insert_list_nat(X,bot_bot_set_list_nat)))),finite_card_list_nat(A3))) ).

% card_Diff1_le
tff(fact_2638_card__Diff1__le,axiom,
    ! [A3: set_set_nat,X: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A3,insert_set_nat(X,bot_bot_set_set_nat)))),finite_card_set_nat(A3))) ).

% card_Diff1_le
tff(fact_2639_card__Diff1__le,axiom,
    ! [A3: set_real,X: real] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(minus_minus_set_real(A3,insert_real(X,bot_bot_set_real)))),finite_card_real(A3))) ).

% card_Diff1_le
tff(fact_2640_card__Diff1__le,axiom,
    ! [A3: set_bool,X: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_bool(minus_minus_set_bool(A3,insert_bool(X,bot_bot_set_bool)))),finite_card_bool(A3))) ).

% card_Diff1_le
tff(fact_2641_card__Diff1__le,axiom,
    ! [A3: set_int,X: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(X,bot_bot_set_int)))),finite_card_int(A3))) ).

% card_Diff1_le
tff(fact_2642_card__Diff1__le,axiom,
    ! [A3: set_nat,X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(X,bot_bot_set_nat)))),aa_set_nat_nat(finite_card_nat,A3))) ).

% card_Diff1_le
tff(fact_2643_finite__induct__select,axiom,
    ! [S: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [T4: set_complex] :
              ( pp(aa_set_complex_bool(ord_less_set_complex(T4),S))
             => ( pp(aa_set_complex_bool(P,T4))
               => ? [X4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(S,T4)))
                    & pp(aa_set_complex_bool(P,insert_complex(X4,T4))) ) ) )
         => pp(aa_set_complex_bool(P,S)) ) ) ) ).

% finite_induct_select
tff(fact_2644_finite__induct__select,axiom,
    ! [S: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S))
     => ( pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat))
       => ( ! [T4: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(T4),S))
             => ( pp(aa_set3135850010578026034t_bool(P,T4))
               => ? [X4: product_prod_nat_nat] :
                    ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,S),T4)))
                    & pp(aa_set3135850010578026034t_bool(P,insert8211810215607154385at_nat(X4,T4))) ) ) )
         => pp(aa_set3135850010578026034t_bool(P,S)) ) ) ) ).

% finite_induct_select
tff(fact_2645_finite__induct__select,axiom,
    ! [S: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [T4: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(T4),S))
             => ( pp(aa_set3347977610044113549t_bool(P,T4))
               => ? [X4: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(S,T4)))
                    & pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(X4,T4))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,S)) ) ) ) ).

% finite_induct_select
tff(fact_2646_finite__induct__select,axiom,
    ! [S: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [T4: set_real] :
              ( pp(aa_set_real_bool(ord_less_set_real(T4),S))
             => ( pp(aa_set_real_bool(P,T4))
               => ? [X4: real] :
                    ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(S,T4)))
                    & pp(aa_set_real_bool(P,insert_real(X4,T4))) ) ) )
         => pp(aa_set_real_bool(P,S)) ) ) ) ).

% finite_induct_select
tff(fact_2647_finite__induct__select,axiom,
    ! [S: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
       => ( ! [T4: set_bool] :
              ( pp(aa_set_bool_bool(ord_less_set_bool(T4),S))
             => ( pp(aa_set_bool_bool(P,T4))
               => ? [X4: bool] :
                    ( pp(aa_set_bool_bool(member_bool(X4),minus_minus_set_bool(S,T4)))
                    & pp(aa_set_bool_bool(P,insert_bool(X4,T4))) ) ) )
         => pp(aa_set_bool_bool(P,S)) ) ) ) ).

% finite_induct_select
tff(fact_2648_finite__induct__select,axiom,
    ! [S: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [T4: set_int] :
              ( pp(aa_set_int_bool(ord_less_set_int(T4),S))
             => ( pp(aa_set_int_bool(P,T4))
               => ? [X4: int] :
                    ( pp(aa_set_int_bool(member_int(X4),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),T4)))
                    & pp(aa_set_int_bool(P,insert_int(X4,T4))) ) ) )
         => pp(aa_set_int_bool(P,S)) ) ) ) ).

% finite_induct_select
tff(fact_2649_finite__induct__select,axiom,
    ! [S: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [T4: set_nat] :
              ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,T4),S))
             => ( pp(aa_set_nat_bool(P,T4))
               => ? [X4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),T4)))
                    & pp(aa_set_nat_bool(P,insert_nat(X4,T4))) ) ) )
         => pp(aa_set_nat_bool(P,S)) ) ) ) ).

% finite_induct_select
tff(fact_2650_psubset__insert__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A3),insert8211810215607154385at_nat(X,B2)))
    <=> ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
         => pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A3),B2)) )
        & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
         => ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
             => pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B2)) )
            & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
             => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),B2)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_2651_psubset__insert__iff,axiom,
    ! [A3: set_set_nat,X: set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A3),insert_set_nat(X,B2)))
    <=> ( ( pp(aa_set_set_nat_bool(member_set_nat(X),B2))
         => pp(aa_set_set_nat_bool(ord_less_set_set_nat(A3),B2)) )
        & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),B2))
         => ( ( pp(aa_set_set_nat_bool(member_set_nat(X),A3))
             => pp(aa_set_set_nat_bool(ord_less_set_set_nat(minus_2163939370556025621et_nat(A3,insert_set_nat(X,bot_bot_set_set_nat))),B2)) )
            & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A3))
             => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),B2)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_2652_psubset__insert__iff,axiom,
    ! [A3: set_real,X: real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A3),insert_real(X,B2)))
    <=> ( ( pp(aa_set_real_bool(member_real(X),B2))
         => pp(aa_set_real_bool(ord_less_set_real(A3),B2)) )
        & ( ~ pp(aa_set_real_bool(member_real(X),B2))
         => ( ( pp(aa_set_real_bool(member_real(X),A3))
             => pp(aa_set_real_bool(ord_less_set_real(minus_minus_set_real(A3,insert_real(X,bot_bot_set_real))),B2)) )
            & ( ~ pp(aa_set_real_bool(member_real(X),A3))
             => pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_2653_psubset__insert__iff,axiom,
    ! [A3: set_bool,X: bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_set_bool(A3),insert_bool(X,B2)))
    <=> ( ( pp(aa_set_bool_bool(member_bool(X),B2))
         => pp(aa_set_bool_bool(ord_less_set_bool(A3),B2)) )
        & ( ~ pp(aa_set_bool_bool(member_bool(X),B2))
         => ( ( pp(aa_set_bool_bool(member_bool(X),A3))
             => pp(aa_set_bool_bool(ord_less_set_bool(minus_minus_set_bool(A3,insert_bool(X,bot_bot_set_bool))),B2)) )
            & ( ~ pp(aa_set_bool_bool(member_bool(X),A3))
             => pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),B2)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_2654_psubset__insert__iff,axiom,
    ! [A3: set_nat,X: nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A3),insert_nat(X,B2)))
    <=> ( ( pp(aa_set_nat_bool(member_nat(X),B2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A3),B2)) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),B2))
         => ( ( pp(aa_set_nat_bool(member_nat(X),A3))
             => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(X,bot_bot_set_nat))),B2)) )
            & ( ~ pp(aa_set_nat_bool(member_nat(X),A3))
             => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_2655_psubset__insert__iff,axiom,
    ! [A3: set_int,X: int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A3),insert_int(X,B2)))
    <=> ( ( pp(aa_set_int_bool(member_int(X),B2))
         => pp(aa_set_int_bool(ord_less_set_int(A3),B2)) )
        & ( ~ pp(aa_set_int_bool(member_int(X),B2))
         => ( ( pp(aa_set_int_bool(member_int(X),A3))
             => pp(aa_set_int_bool(ord_less_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(X,bot_bot_set_int))),B2)) )
            & ( ~ pp(aa_set_int_bool(member_int(X),A3))
             => pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_2656_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A: bool,B: bool,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A,B),X5)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3),X5) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_2657_card_Oremove,axiom,
    ! [A3: set_list_nat,X: list_nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( member_list_nat(X,A3)
       => ( finite_card_list_nat(A3) = aa_nat_nat(suc,finite_card_list_nat(minus_7954133019191499631st_nat(A3,insert_list_nat(X,bot_bot_set_list_nat)))) ) ) ) ).

% card.remove
tff(fact_2658_card_Oremove,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X),A3))
       => ( finite_card_set_nat(A3) = aa_nat_nat(suc,finite_card_set_nat(minus_2163939370556025621et_nat(A3,insert_set_nat(X,bot_bot_set_set_nat)))) ) ) ) ).

% card.remove
tff(fact_2659_card_Oremove,axiom,
    ! [A3: set_complex,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(member_complex(X),A3))
       => ( finite_card_complex(A3) = aa_nat_nat(suc,finite_card_complex(minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% card.remove
tff(fact_2660_card_Oremove,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
       => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3) = aa_nat_nat(suc,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)))) ) ) ) ).

% card.remove
tff(fact_2661_card_Oremove,axiom,
    ! [A3: set_Extended_enat,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A3))
       => ( finite121521170596916366d_enat(A3) = aa_nat_nat(suc,finite121521170596916366d_enat(minus_925952699566721837d_enat(A3,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% card.remove
tff(fact_2662_card_Oremove,axiom,
    ! [A3: set_real,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(member_real(X),A3))
       => ( finite_card_real(A3) = aa_nat_nat(suc,finite_card_real(minus_minus_set_real(A3,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% card.remove
tff(fact_2663_card_Oremove,axiom,
    ! [A3: set_bool,X: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( pp(aa_set_bool_bool(member_bool(X),A3))
       => ( finite_card_bool(A3) = aa_nat_nat(suc,finite_card_bool(minus_minus_set_bool(A3,insert_bool(X,bot_bot_set_bool)))) ) ) ) ).

% card.remove
tff(fact_2664_card_Oremove,axiom,
    ! [A3: set_int,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(member_int(X),A3))
       => ( finite_card_int(A3) = aa_nat_nat(suc,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(X,bot_bot_set_int)))) ) ) ) ).

% card.remove
tff(fact_2665_card_Oremove,axiom,
    ! [A3: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(member_nat(X),A3))
       => ( aa_set_nat_nat(finite_card_nat,A3) = aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(X,bot_bot_set_nat)))) ) ) ) ).

% card.remove
tff(fact_2666_card_Oinsert__remove,axiom,
    ! [A3: set_list_nat,X: list_nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( finite_card_list_nat(insert_list_nat(X,A3)) = aa_nat_nat(suc,finite_card_list_nat(minus_7954133019191499631st_nat(A3,insert_list_nat(X,bot_bot_set_list_nat)))) ) ) ).

% card.insert_remove
tff(fact_2667_card_Oinsert__remove,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( finite_card_set_nat(insert_set_nat(X,A3)) = aa_nat_nat(suc,finite_card_set_nat(minus_2163939370556025621et_nat(A3,insert_set_nat(X,bot_bot_set_set_nat)))) ) ) ).

% card.insert_remove
tff(fact_2668_card_Oinsert__remove,axiom,
    ! [A3: set_complex,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( finite_card_complex(insert_complex(X,A3)) = aa_nat_nat(suc,finite_card_complex(minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% card.insert_remove
tff(fact_2669_card_Oinsert__remove,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,insert8211810215607154385at_nat(X,A3)) = aa_nat_nat(suc,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)))) ) ) ).

% card.insert_remove
tff(fact_2670_card_Oinsert__remove,axiom,
    ! [A3: set_Extended_enat,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( finite121521170596916366d_enat(insert_Extended_enat(X,A3)) = aa_nat_nat(suc,finite121521170596916366d_enat(minus_925952699566721837d_enat(A3,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ).

% card.insert_remove
tff(fact_2671_card_Oinsert__remove,axiom,
    ! [A3: set_real,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( finite_card_real(insert_real(X,A3)) = aa_nat_nat(suc,finite_card_real(minus_minus_set_real(A3,insert_real(X,bot_bot_set_real)))) ) ) ).

% card.insert_remove
tff(fact_2672_card_Oinsert__remove,axiom,
    ! [A3: set_bool,X: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( finite_card_bool(insert_bool(X,A3)) = aa_nat_nat(suc,finite_card_bool(minus_minus_set_bool(A3,insert_bool(X,bot_bot_set_bool)))) ) ) ).

% card.insert_remove
tff(fact_2673_card_Oinsert__remove,axiom,
    ! [A3: set_int,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( finite_card_int(insert_int(X,A3)) = aa_nat_nat(suc,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(X,bot_bot_set_int)))) ) ) ).

% card.insert_remove
tff(fact_2674_card_Oinsert__remove,axiom,
    ! [A3: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( aa_set_nat_nat(finite_card_nat,insert_nat(X,A3)) = aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(X,bot_bot_set_nat)))) ) ) ).

% card.insert_remove
tff(fact_2675_card__Suc__Diff1,axiom,
    ! [A3: set_list_nat,X: list_nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( member_list_nat(X,A3)
       => ( aa_nat_nat(suc,finite_card_list_nat(minus_7954133019191499631st_nat(A3,insert_list_nat(X,bot_bot_set_list_nat)))) = finite_card_list_nat(A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_2676_card__Suc__Diff1,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X),A3))
       => ( aa_nat_nat(suc,finite_card_set_nat(minus_2163939370556025621et_nat(A3,insert_set_nat(X,bot_bot_set_set_nat)))) = finite_card_set_nat(A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_2677_card__Suc__Diff1,axiom,
    ! [A3: set_complex,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(member_complex(X),A3))
       => ( aa_nat_nat(suc,finite_card_complex(minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex)))) = finite_card_complex(A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_2678_card__Suc__Diff1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
       => ( aa_nat_nat(suc,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)))) = aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_2679_card__Suc__Diff1,axiom,
    ! [A3: set_Extended_enat,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A3))
       => ( aa_nat_nat(suc,finite121521170596916366d_enat(minus_925952699566721837d_enat(A3,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) = finite121521170596916366d_enat(A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_2680_card__Suc__Diff1,axiom,
    ! [A3: set_real,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(member_real(X),A3))
       => ( aa_nat_nat(suc,finite_card_real(minus_minus_set_real(A3,insert_real(X,bot_bot_set_real)))) = finite_card_real(A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_2681_card__Suc__Diff1,axiom,
    ! [A3: set_bool,X: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( pp(aa_set_bool_bool(member_bool(X),A3))
       => ( aa_nat_nat(suc,finite_card_bool(minus_minus_set_bool(A3,insert_bool(X,bot_bot_set_bool)))) = finite_card_bool(A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_2682_card__Suc__Diff1,axiom,
    ! [A3: set_int,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(member_int(X),A3))
       => ( aa_nat_nat(suc,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(X,bot_bot_set_int)))) = finite_card_int(A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_2683_card__Suc__Diff1,axiom,
    ! [A3: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(member_nat(X),A3))
       => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(X,bot_bot_set_nat)))) = aa_set_nat_nat(finite_card_nat,A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_2684_card__Diff1__less,axiom,
    ! [A3: set_list_nat,X: list_nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( member_list_nat(X,A3)
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A3,insert_list_nat(X,bot_bot_set_list_nat)))),finite_card_list_nat(A3))) ) ) ).

% card_Diff1_less
tff(fact_2685_card__Diff1__less,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X),A3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A3,insert_set_nat(X,bot_bot_set_set_nat)))),finite_card_set_nat(A3))) ) ) ).

% card_Diff1_less
tff(fact_2686_card__Diff1__less,axiom,
    ! [A3: set_complex,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(member_complex(X),A3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_complex(minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex)))),finite_card_complex(A3))) ) ) ).

% card_Diff1_less
tff(fact_2687_card__Diff1__less,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)))),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3))) ) ) ).

% card_Diff1_less
tff(fact_2688_card__Diff1__less,axiom,
    ! [A3: set_Extended_enat,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite121521170596916366d_enat(minus_925952699566721837d_enat(A3,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))),finite121521170596916366d_enat(A3))) ) ) ).

% card_Diff1_less
tff(fact_2689_card__Diff1__less,axiom,
    ! [A3: set_real,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(member_real(X),A3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_real(minus_minus_set_real(A3,insert_real(X,bot_bot_set_real)))),finite_card_real(A3))) ) ) ).

% card_Diff1_less
tff(fact_2690_card__Diff1__less,axiom,
    ! [A3: set_bool,X: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( pp(aa_set_bool_bool(member_bool(X),A3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_bool(minus_minus_set_bool(A3,insert_bool(X,bot_bot_set_bool)))),finite_card_bool(A3))) ) ) ).

% card_Diff1_less
tff(fact_2691_card__Diff1__less,axiom,
    ! [A3: set_int,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(member_int(X),A3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(X,bot_bot_set_int)))),finite_card_int(A3))) ) ) ).

% card_Diff1_less
tff(fact_2692_card__Diff1__less,axiom,
    ! [A3: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(member_nat(X),A3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(X,bot_bot_set_nat)))),aa_set_nat_nat(finite_card_nat,A3))) ) ) ).

% card_Diff1_less
tff(fact_2693_card__Diff2__less,axiom,
    ! [A3: set_list_nat,X: list_nat,Y: list_nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( member_list_nat(X,A3)
       => ( member_list_nat(Y,A3)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_list_nat(minus_7954133019191499631st_nat(minus_7954133019191499631st_nat(A3,insert_list_nat(X,bot_bot_set_list_nat)),insert_list_nat(Y,bot_bot_set_list_nat)))),finite_card_list_nat(A3))) ) ) ) ).

% card_Diff2_less
tff(fact_2694_card__Diff2__less,axiom,
    ! [A3: set_set_nat,X: set_nat,Y: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X),A3))
       => ( pp(aa_set_set_nat_bool(member_set_nat(Y),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_set_nat(minus_2163939370556025621et_nat(minus_2163939370556025621et_nat(A3,insert_set_nat(X,bot_bot_set_set_nat)),insert_set_nat(Y,bot_bot_set_set_nat)))),finite_card_set_nat(A3))) ) ) ) ).

% card_Diff2_less
tff(fact_2695_card__Diff2__less,axiom,
    ! [A3: set_complex,X: complex,Y: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(member_complex(X),A3))
       => ( pp(aa_set_complex_bool(member_complex(Y),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_complex(minus_811609699411566653omplex(minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex)),insert_complex(Y,bot_bot_set_complex)))),finite_card_complex(A3))) ) ) ) ).

% card_Diff2_less
tff(fact_2696_card__Diff2__less,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
       => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),insert8211810215607154385at_nat(Y,bot_bo2099793752762293965at_nat)))),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3))) ) ) ) ).

% card_Diff2_less
tff(fact_2697_card__Diff2__less,axiom,
    ! [A3: set_Extended_enat,X: extended_enat,Y: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A3))
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite121521170596916366d_enat(minus_925952699566721837d_enat(minus_925952699566721837d_enat(A3,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)),insert_Extended_enat(Y,bot_bo7653980558646680370d_enat)))),finite121521170596916366d_enat(A3))) ) ) ) ).

% card_Diff2_less
tff(fact_2698_card__Diff2__less,axiom,
    ! [A3: set_real,X: real,Y: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(member_real(X),A3))
       => ( pp(aa_set_real_bool(member_real(Y),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_real(minus_minus_set_real(minus_minus_set_real(A3,insert_real(X,bot_bot_set_real)),insert_real(Y,bot_bot_set_real)))),finite_card_real(A3))) ) ) ) ).

% card_Diff2_less
tff(fact_2699_card__Diff2__less,axiom,
    ! [A3: set_bool,X: bool,Y: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( pp(aa_set_bool_bool(member_bool(X),A3))
       => ( pp(aa_set_bool_bool(member_bool(Y),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_bool(minus_minus_set_bool(minus_minus_set_bool(A3,insert_bool(X,bot_bot_set_bool)),insert_bool(Y,bot_bot_set_bool)))),finite_card_bool(A3))) ) ) ) ).

% card_Diff2_less
tff(fact_2700_card__Diff2__less,axiom,
    ! [A3: set_int,X: int,Y: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(member_int(X),A3))
       => ( pp(aa_set_int_bool(member_int(Y),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(X,bot_bot_set_int))),insert_int(Y,bot_bot_set_int)))),finite_card_int(A3))) ) ) ) ).

% card_Diff2_less
tff(fact_2701_card__Diff2__less,axiom,
    ! [A3: set_nat,X: nat,Y: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(member_nat(X),A3))
       => ( pp(aa_set_nat_bool(member_nat(Y),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(X,bot_bot_set_nat))),insert_nat(Y,bot_bot_set_nat)))),aa_set_nat_nat(finite_card_nat,A3))) ) ) ) ).

% card_Diff2_less
tff(fact_2702_card__Diff1__less__iff,axiom,
    ! [A3: set_list_nat,X: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A3,insert_list_nat(X,bot_bot_set_list_nat)))),finite_card_list_nat(A3)))
    <=> ( finite8100373058378681591st_nat(A3)
        & member_list_nat(X,A3) ) ) ).

% card_Diff1_less_iff
tff(fact_2703_card__Diff1__less__iff,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A3,insert_set_nat(X,bot_bot_set_set_nat)))),finite_card_set_nat(A3)))
    <=> ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
        & pp(aa_set_set_nat_bool(member_set_nat(X),A3)) ) ) ).

% card_Diff1_less_iff
tff(fact_2704_card__Diff1__less__iff,axiom,
    ! [A3: set_complex,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_complex(minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex)))),finite_card_complex(A3)))
    <=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
        & pp(aa_set_complex_bool(member_complex(X),A3)) ) ) ).

% card_Diff1_less_iff
tff(fact_2705_card__Diff1__less__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)))),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)))
    <=> ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
        & pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3)) ) ) ).

% card_Diff1_less_iff
tff(fact_2706_card__Diff1__less__iff,axiom,
    ! [A3: set_Extended_enat,X: extended_enat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite121521170596916366d_enat(minus_925952699566721837d_enat(A3,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))),finite121521170596916366d_enat(A3)))
    <=> ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
        & pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A3)) ) ) ).

% card_Diff1_less_iff
tff(fact_2707_card__Diff1__less__iff,axiom,
    ! [A3: set_real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_real(minus_minus_set_real(A3,insert_real(X,bot_bot_set_real)))),finite_card_real(A3)))
    <=> ( pp(aa_set_real_bool(finite_finite_real,A3))
        & pp(aa_set_real_bool(member_real(X),A3)) ) ) ).

% card_Diff1_less_iff
tff(fact_2708_card__Diff1__less__iff,axiom,
    ! [A3: set_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_bool(minus_minus_set_bool(A3,insert_bool(X,bot_bot_set_bool)))),finite_card_bool(A3)))
    <=> ( pp(aa_set_bool_bool(finite_finite_bool,A3))
        & pp(aa_set_bool_bool(member_bool(X),A3)) ) ) ).

% card_Diff1_less_iff
tff(fact_2709_card__Diff1__less__iff,axiom,
    ! [A3: set_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(X,bot_bot_set_int)))),finite_card_int(A3)))
    <=> ( pp(aa_set_int_bool(finite_finite_int,A3))
        & pp(aa_set_int_bool(member_int(X),A3)) ) ) ).

% card_Diff1_less_iff
tff(fact_2710_card__Diff1__less__iff,axiom,
    ! [A3: set_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(X,bot_bot_set_nat)))),aa_set_nat_nat(finite_card_nat,A3)))
    <=> ( pp(aa_set_nat_bool(finite_finite_nat,A3))
        & pp(aa_set_nat_bool(member_nat(X),A3)) ) ) ).

% card_Diff1_less_iff
tff(fact_2711_card__Diff__singleton,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
     => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_2712_card__Diff__singleton,axiom,
    ! [X: complex,A3: set_complex] :
      ( pp(aa_set_complex_bool(member_complex(X),A3))
     => ( finite_card_complex(minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A3)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_2713_card__Diff__singleton,axiom,
    ! [X: list_nat,A3: set_list_nat] :
      ( member_list_nat(X,A3)
     => ( finite_card_list_nat(minus_7954133019191499631st_nat(A3,insert_list_nat(X,bot_bot_set_list_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A3)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_2714_card__Diff__singleton,axiom,
    ! [X: set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),A3))
     => ( finite_card_set_nat(minus_2163939370556025621et_nat(A3,insert_set_nat(X,bot_bot_set_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A3)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_2715_card__Diff__singleton,axiom,
    ! [X: real,A3: set_real] :
      ( pp(aa_set_real_bool(member_real(X),A3))
     => ( finite_card_real(minus_minus_set_real(A3,insert_real(X,bot_bot_set_real))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_real(A3)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_2716_card__Diff__singleton,axiom,
    ! [X: bool,A3: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(X),A3))
     => ( finite_card_bool(minus_minus_set_bool(A3,insert_bool(X,bot_bot_set_bool))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_bool(A3)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_2717_card__Diff__singleton,axiom,
    ! [X: int,A3: set_int] :
      ( pp(aa_set_int_bool(member_int(X),A3))
     => ( finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(X,bot_bot_set_int))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A3)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_2718_card__Diff__singleton,axiom,
    ! [X: nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),A3))
     => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(X,bot_bot_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,A3)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_2719_card__Diff__singleton__if,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
       => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),one_one_nat) ) )
      & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
       => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))) = aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3) ) ) ) ).

% card_Diff_singleton_if
tff(fact_2720_card__Diff__singleton__if,axiom,
    ! [X: complex,A3: set_complex] :
      ( ( pp(aa_set_complex_bool(member_complex(X),A3))
       => ( finite_card_complex(minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A3)),one_one_nat) ) )
      & ( ~ pp(aa_set_complex_bool(member_complex(X),A3))
       => ( finite_card_complex(minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex))) = finite_card_complex(A3) ) ) ) ).

% card_Diff_singleton_if
tff(fact_2721_card__Diff__singleton__if,axiom,
    ! [X: list_nat,A3: set_list_nat] :
      ( ( member_list_nat(X,A3)
       => ( finite_card_list_nat(minus_7954133019191499631st_nat(A3,insert_list_nat(X,bot_bot_set_list_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A3)),one_one_nat) ) )
      & ( ~ member_list_nat(X,A3)
       => ( finite_card_list_nat(minus_7954133019191499631st_nat(A3,insert_list_nat(X,bot_bot_set_list_nat))) = finite_card_list_nat(A3) ) ) ) ).

% card_Diff_singleton_if
tff(fact_2722_card__Diff__singleton__if,axiom,
    ! [X: set_nat,A3: set_set_nat] :
      ( ( pp(aa_set_set_nat_bool(member_set_nat(X),A3))
       => ( finite_card_set_nat(minus_2163939370556025621et_nat(A3,insert_set_nat(X,bot_bot_set_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A3)),one_one_nat) ) )
      & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A3))
       => ( finite_card_set_nat(minus_2163939370556025621et_nat(A3,insert_set_nat(X,bot_bot_set_set_nat))) = finite_card_set_nat(A3) ) ) ) ).

% card_Diff_singleton_if
tff(fact_2723_card__Diff__singleton__if,axiom,
    ! [X: real,A3: set_real] :
      ( ( pp(aa_set_real_bool(member_real(X),A3))
       => ( finite_card_real(minus_minus_set_real(A3,insert_real(X,bot_bot_set_real))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_real(A3)),one_one_nat) ) )
      & ( ~ pp(aa_set_real_bool(member_real(X),A3))
       => ( finite_card_real(minus_minus_set_real(A3,insert_real(X,bot_bot_set_real))) = finite_card_real(A3) ) ) ) ).

% card_Diff_singleton_if
tff(fact_2724_card__Diff__singleton__if,axiom,
    ! [X: bool,A3: set_bool] :
      ( ( pp(aa_set_bool_bool(member_bool(X),A3))
       => ( finite_card_bool(minus_minus_set_bool(A3,insert_bool(X,bot_bot_set_bool))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_bool(A3)),one_one_nat) ) )
      & ( ~ pp(aa_set_bool_bool(member_bool(X),A3))
       => ( finite_card_bool(minus_minus_set_bool(A3,insert_bool(X,bot_bot_set_bool))) = finite_card_bool(A3) ) ) ) ).

% card_Diff_singleton_if
tff(fact_2725_card__Diff__singleton__if,axiom,
    ! [X: int,A3: set_int] :
      ( ( pp(aa_set_int_bool(member_int(X),A3))
       => ( finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(X,bot_bot_set_int))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A3)),one_one_nat) ) )
      & ( ~ pp(aa_set_int_bool(member_int(X),A3))
       => ( finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(X,bot_bot_set_int))) = finite_card_int(A3) ) ) ) ).

% card_Diff_singleton_if
tff(fact_2726_card__Diff__singleton__if,axiom,
    ! [X: nat,A3: set_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(X),A3))
       => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(X,bot_bot_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,A3)),one_one_nat) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(X),A3))
       => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(X,bot_bot_set_nat))) = aa_set_nat_nat(finite_card_nat,A3) ) ) ) ).

% card_Diff_singleton_if
tff(fact_2727_enumerate__Suc_H,axiom,
    ! [S: set_nat,N: nat] : aa_nat_nat(infini8530281810654367211te_nat(S),aa_nat_nat(suc,N)) = aa_nat_nat(infini8530281810654367211te_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat(aa_nat_nat(infini8530281810654367211te_nat(S),zero_zero_nat),bot_bot_set_nat))),N) ).

% enumerate_Suc'
tff(fact_2728_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,insert8211810215607154385at_nat(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_2729_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(insert_real(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_2730_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_bool(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_bool(insert_bool(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_2731_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_complex,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(insert_complex(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_2732_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_list_nat,X: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(insert_list_nat(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_2733_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_set_nat,X: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(insert_set_nat(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_2734_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,insert_nat(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_2735_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(insert_int(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_2736_vebt__pred_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),zero_zero_nat)
     => ( ! [A: bool,Uw2: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(A,Uw2),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A: bool,B: bool,Va2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Vb2)
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2),Vf2)
             => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Vj2)
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2),X5) ) ) ) ) ) ) ).

% vebt_pred.cases
tff(fact_2737_vebt__succ_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,B: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,B),zero_zero_nat)
     => ( ! [Uv2: bool,Uw2: bool,N2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uv2,Uw2),aa_nat_nat(suc,N2))
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Va3)
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Ve2)
           => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Vi2)
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2),X5) ) ) ) ) ) ).

% vebt_succ.cases
tff(fact_2738_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool,Uw2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),X5)
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2),X5)
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),X5) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_2739_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A: bool,B: bool,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A,B),X5)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),X5)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),X5)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X5)
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2),X5) ) ) ) ) ).

% vebt_member.cases
tff(fact_2740_vebt__delete_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A: bool,B: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(A,B),zero_zero_nat)
     => ( ! [A: bool,B: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(A,B),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A: bool,B: bool,N2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,N2)))
         => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2),Uu2)
           => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2,Smry2),X5)
             => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2),X5)
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2),X5) ) ) ) ) ) ) ).

% vebt_delete.cases
tff(fact_2741_vebt__insert_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A: bool,B: bool,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A,B),X5)
     => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info,zero_zero_nat,Ts,S3),X5)
       => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S3),X5)
         => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary2),X5)
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2),X5) ) ) ) ) ).

% vebt_insert.cases
tff(fact_2742_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : finite5795047828879050333T_VEBT(set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_2743_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : pp(aa_set_nat_bool(finite_finite_nat,set_nat2(Xs))) ).

% List.finite_set
tff(fact_2744_List_Ofinite__set,axiom,
    ! [Xs: list_int] : pp(aa_set_int_bool(finite_finite_int,set_int2(Xs))) ).

% List.finite_set
tff(fact_2745_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : pp(aa_set_complex_bool(finite3207457112153483333omplex,set_complex2(Xs))) ).

% List.finite_set
tff(fact_2746_List_Ofinite__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat] : pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,set_Pr5648618587558075414at_nat(Xs))) ).

% List.finite_set
tff(fact_2747_List_Ofinite__set,axiom,
    ! [Xs: list_Extended_enat] : pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,set_Extended_enat2(Xs))) ).

% List.finite_set
tff(fact_2748_Bolzano,axiom,
    ! [A2: real,B3: real,P: fun_re1255401033344146673l_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( ! [A: real,B: real,C3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B),C3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C3))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),C3)) ) ) ) )
       => ( ! [X5: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X5))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),B3))
               => ? [D3: real] :
                    ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D3))
                    & ! [A: real,B: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X5))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),B))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),D3)) )
                       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B)) ) ) ) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A2),B3)) ) ) ) ).

% Bolzano
tff(fact_2749_card__length,axiom,
    ! [Xs: list_complex] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(set_complex2(Xs))),size_s3451745648224563538omplex(Xs))) ).

% card_length
tff(fact_2750_card__length,axiom,
    ! [Xs: list_list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(set_list_nat2(Xs))),size_s3023201423986296836st_nat(Xs))) ).

% card_length
tff(fact_2751_card__length,axiom,
    ! [Xs: list_set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(set_set_nat2(Xs))),size_s3254054031482475050et_nat(Xs))) ).

% card_length
tff(fact_2752_card__length,axiom,
    ! [Xs: list_int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(set_int2(Xs))),size_size_list_int(Xs))) ).

% card_length
tff(fact_2753_card__length,axiom,
    ! [Xs: list_VEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite7802652506058667612T_VEBT(set_VEBT_VEBT2(Xs))),size_s6755466524823107622T_VEBT(Xs))) ).

% card_length
tff(fact_2754_card__length,axiom,
    ! [Xs: list_bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_bool(set_bool2(Xs))),size_size_list_bool(Xs))) ).

% card_length
tff(fact_2755_card__length,axiom,
    ! [Xs: list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,set_nat2(Xs))),size_size_list_nat(Xs))) ).

% card_length
tff(fact_2756_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_real(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2757_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2758_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_int(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2759_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2760_length__pos__if__in__set,axiom,
    ! [X: bool,Xs: list_bool] :
      ( pp(aa_set_bool_bool(member_bool(X),set_bool2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_bool(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2761_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2762_the__elem__eq,axiom,
    ! [X: product_prod_nat_nat] : the_el2281957884133575798at_nat(insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) = X ).

% the_elem_eq
tff(fact_2763_the__elem__eq,axiom,
    ! [X: real] : the_elem_real(insert_real(X,bot_bot_set_real)) = X ).

% the_elem_eq
tff(fact_2764_the__elem__eq,axiom,
    ! [X: bool] :
      ( pp(aa_set_bool_bool(the_elem_bool,insert_bool(X,bot_bot_set_bool)))
    <=> pp(X) ) ).

% the_elem_eq
tff(fact_2765_the__elem__eq,axiom,
    ! [X: nat] : aa_set_nat_nat(the_elem_nat,insert_nat(X,bot_bot_set_nat)) = X ).

% the_elem_eq
tff(fact_2766_the__elem__eq,axiom,
    ! [X: int] : the_elem_int(insert_int(X,bot_bot_set_int)) = X ).

% the_elem_eq
tff(fact_2767_case4_I1_J,axiom,
    ! [X4: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(treeList2)))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(X4),na))
        & ! [Xa: vEBT_VEBT] :
            ( pp(aa_nat_bool(vEBT_invar_vebt(Xa),na))
           => ( ( vEBT_VEBT_set_vebt(X4) = vEBT_VEBT_set_vebt(Xa) )
             => ( Xa = X4 ) ) ) ) ) ).

% case4(1)
tff(fact_2768_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),X))
       => ( ! [A: bool,B: bool] :
              ( ( X = vEBT_Leaf(A,B) )
             => ( ( ( pp(B)
                   => ( Y = some_nat(one_one_nat) ) )
                  & ( ~ pp(B)
                   => ( ( pp(A)
                       => ( Y = some_nat(zero_zero_nat) ) )
                      & ( ~ pp(A)
                       => ( Y = none_nat ) ) ) ) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Leaf(A,B))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = some_nat(Ma2) )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2))) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_2769_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),X))
       => ( ! [A: bool,B: bool] :
              ( ( X = vEBT_Leaf(A,B) )
             => ( ( ( pp(A)
                   => ( Y = some_nat(zero_zero_nat) ) )
                  & ( ~ pp(A)
                   => ( ( pp(B)
                       => ( Y = some_nat(one_one_nat) ) )
                      & ( ~ pp(B)
                       => ( Y = none_nat ) ) ) ) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf(A,B))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = some_nat(Mi2) )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2))) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_2770_is__singletonI,axiom,
    ! [X: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(is_sin2850979758926227957at_nat,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))) ).

% is_singletonI
tff(fact_2771_is__singletonI,axiom,
    ! [X: real] : pp(aa_set_real_bool(is_singleton_real,insert_real(X,bot_bot_set_real))) ).

% is_singletonI
tff(fact_2772_is__singletonI,axiom,
    ! [X: bool] : pp(aa_set_bool_bool(is_singleton_bool,insert_bool(X,bot_bot_set_bool))) ).

% is_singletonI
tff(fact_2773_is__singletonI,axiom,
    ! [X: nat] : pp(aa_set_nat_bool(is_singleton_nat,insert_nat(X,bot_bot_set_nat))) ).

% is_singletonI
tff(fact_2774_is__singletonI,axiom,
    ! [X: int] : pp(aa_set_int_bool(is_singleton_int,insert_int(X,bot_bot_set_int))) ).

% is_singletonI
tff(fact_2775_case4_I5_J,axiom,
    m = aa_nat_nat(suc,na) ).

% case4(5)
tff(fact_2776_is__singleton__the__elem,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(is_sin2850979758926227957at_nat,A3))
    <=> ( A3 = insert8211810215607154385at_nat(the_el2281957884133575798at_nat(A3),bot_bo2099793752762293965at_nat) ) ) ).

% is_singleton_the_elem
tff(fact_2777_is__singleton__the__elem,axiom,
    ! [A3: set_real] :
      ( pp(aa_set_real_bool(is_singleton_real,A3))
    <=> ( A3 = insert_real(the_elem_real(A3),bot_bot_set_real) ) ) ).

% is_singleton_the_elem
tff(fact_2778_is__singleton__the__elem,axiom,
    ! [A3: set_bool] :
      ( pp(aa_set_bool_bool(is_singleton_bool,A3))
    <=> ( A3 = insert_bool(aa_set_bool_bool(the_elem_bool,A3),bot_bot_set_bool) ) ) ).

% is_singleton_the_elem
tff(fact_2779_is__singleton__the__elem,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(is_singleton_nat,A3))
    <=> ( A3 = insert_nat(aa_set_nat_nat(the_elem_nat,A3),bot_bot_set_nat) ) ) ).

% is_singleton_the_elem
tff(fact_2780_is__singleton__the__elem,axiom,
    ! [A3: set_int] :
      ( pp(aa_set_int_bool(is_singleton_int,A3))
    <=> ( A3 = insert_int(the_elem_int(A3),bot_bot_set_int) ) ) ).

% is_singleton_the_elem
tff(fact_2781_is__singletonI_H,axiom,
    ! [A3: set_set_nat] :
      ( ( A3 != bot_bot_set_set_nat )
     => ( ! [X5: set_nat,Y3: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X5),A3))
           => ( pp(aa_set_set_nat_bool(member_set_nat(Y3),A3))
             => ( X5 = Y3 ) ) )
       => pp(aa_set_set_nat_bool(is_singleton_set_nat,A3)) ) ) ).

% is_singletonI'
tff(fact_2782_is__singletonI_H,axiom,
    ! [A3: set_real] :
      ( ( A3 != bot_bot_set_real )
     => ( ! [X5: real,Y3: real] :
            ( pp(aa_set_real_bool(member_real(X5),A3))
           => ( pp(aa_set_real_bool(member_real(Y3),A3))
             => ( X5 = Y3 ) ) )
       => pp(aa_set_real_bool(is_singleton_real,A3)) ) ) ).

% is_singletonI'
tff(fact_2783_is__singletonI_H,axiom,
    ! [A3: set_bool] :
      ( ( A3 != bot_bot_set_bool )
     => ( ! [X5: bool,Y3: bool] :
            ( pp(aa_set_bool_bool(member_bool(X5),A3))
           => ( pp(aa_set_bool_bool(member_bool(Y3),A3))
             => ( pp(X5)
              <=> pp(Y3) ) ) )
       => pp(aa_set_bool_bool(is_singleton_bool,A3)) ) ) ).

% is_singletonI'
tff(fact_2784_is__singletonI_H,axiom,
    ! [A3: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [X5: nat,Y3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A3))
           => ( pp(aa_set_nat_bool(member_nat(Y3),A3))
             => ( X5 = Y3 ) ) )
       => pp(aa_set_nat_bool(is_singleton_nat,A3)) ) ) ).

% is_singletonI'
tff(fact_2785_is__singletonI_H,axiom,
    ! [A3: set_int] :
      ( ( A3 != bot_bot_set_int )
     => ( ! [X5: int,Y3: int] :
            ( pp(aa_set_int_bool(member_int(X5),A3))
           => ( pp(aa_set_int_bool(member_int(Y3),A3))
             => ( X5 = Y3 ) ) )
       => pp(aa_set_int_bool(is_singleton_int,A3)) ) ) ).

% is_singletonI'
tff(fact_2786_length__induct,axiom,
    ! [P: fun_li6006399982103069984T_bool,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys: list_VEBT_VEBT] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(Ys)),size_s6755466524823107622T_VEBT(Xs2)))
             => pp(aa_lis2822252094023462681T_bool(P,Ys)) )
         => pp(aa_lis2822252094023462681T_bool(P,Xs2)) )
     => pp(aa_lis2822252094023462681T_bool(P,Xs)) ) ).

% length_induct
tff(fact_2787_length__induct,axiom,
    ! [P: fun_list_bool_bool,Xs: list_bool] :
      ( ! [Xs2: list_bool] :
          ( ! [Ys: list_bool] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(Ys)),size_size_list_bool(Xs2)))
             => pp(aa_list_bool_bool(P,Ys)) )
         => pp(aa_list_bool_bool(P,Xs2)) )
     => pp(aa_list_bool_bool(P,Xs)) ) ).

% length_induct
tff(fact_2788_length__induct,axiom,
    ! [P: fun_list_nat_bool,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys: list_nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(Ys)),size_size_list_nat(Xs2)))
             => pp(aa_list_nat_bool(P,Ys)) )
         => pp(aa_list_nat_bool(P,Xs2)) )
     => pp(aa_list_nat_bool(P,Xs)) ) ).

% length_induct
tff(fact_2789_finite__maxlen,axiom,
    ! [M5: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M5)
     => ? [N2: nat] :
        ! [X4: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X4,M5)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(X4)),N2)) ) ) ).

% finite_maxlen
tff(fact_2790_finite__maxlen,axiom,
    ! [M5: set_list_bool] :
      ( finite7866827932528546246t_bool(M5)
     => ? [N2: nat] :
        ! [X4: list_bool] :
          ( member_list_bool(X4,M5)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(X4)),N2)) ) ) ).

% finite_maxlen
tff(fact_2791_finite__maxlen,axiom,
    ! [M5: set_list_nat] :
      ( finite8100373058378681591st_nat(M5)
     => ? [N2: nat] :
        ! [X4: list_nat] :
          ( member_list_nat(X4,M5)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(X4)),N2)) ) ) ).

% finite_maxlen
tff(fact_2792_is__singleton__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(is_sin2850979758926227957at_nat,A3))
    <=> ? [X3: product_prod_nat_nat] : A3 = insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat) ) ).

% is_singleton_def
tff(fact_2793_is__singleton__def,axiom,
    ! [A3: set_real] :
      ( pp(aa_set_real_bool(is_singleton_real,A3))
    <=> ? [X3: real] : A3 = insert_real(X3,bot_bot_set_real) ) ).

% is_singleton_def
tff(fact_2794_is__singleton__def,axiom,
    ! [A3: set_bool] :
      ( pp(aa_set_bool_bool(is_singleton_bool,A3))
    <=> ? [X3: bool] : A3 = insert_bool(X3,bot_bot_set_bool) ) ).

% is_singleton_def
tff(fact_2795_is__singleton__def,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(is_singleton_nat,A3))
    <=> ? [X3: nat] : A3 = insert_nat(X3,bot_bot_set_nat) ) ).

% is_singleton_def
tff(fact_2796_is__singleton__def,axiom,
    ! [A3: set_int] :
      ( pp(aa_set_int_bool(is_singleton_int,A3))
    <=> ? [X3: int] : A3 = insert_int(X3,bot_bot_set_int) ) ).

% is_singleton_def
tff(fact_2797_is__singletonE,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(is_sin2850979758926227957at_nat,A3))
     => ~ ! [X5: product_prod_nat_nat] : A3 != insert8211810215607154385at_nat(X5,bot_bo2099793752762293965at_nat) ) ).

% is_singletonE
tff(fact_2798_is__singletonE,axiom,
    ! [A3: set_real] :
      ( pp(aa_set_real_bool(is_singleton_real,A3))
     => ~ ! [X5: real] : A3 != insert_real(X5,bot_bot_set_real) ) ).

% is_singletonE
tff(fact_2799_is__singletonE,axiom,
    ! [A3: set_bool] :
      ( pp(aa_set_bool_bool(is_singleton_bool,A3))
     => ~ ! [X5: bool] : A3 != insert_bool(X5,bot_bot_set_bool) ) ).

% is_singletonE
tff(fact_2800_is__singletonE,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(is_singleton_nat,A3))
     => ~ ! [X5: nat] : A3 != insert_nat(X5,bot_bot_set_nat) ) ).

% is_singletonE
tff(fact_2801_is__singletonE,axiom,
    ! [A3: set_int] :
      ( pp(aa_set_int_bool(is_singleton_int,A3))
     => ~ ! [X5: int] : A3 != insert_int(X5,bot_bot_set_int) ) ).

% is_singletonE
tff(fact_2802_is__singleton__altdef,axiom,
    ! [A3: set_complex] :
      ( pp(aa_set_complex_bool(is_singleton_complex,A3))
    <=> ( finite_card_complex(A3) = one_one_nat ) ) ).

% is_singleton_altdef
tff(fact_2803_is__singleton__altdef,axiom,
    ! [A3: set_list_nat] :
      ( is_sin2641923865335537900st_nat(A3)
    <=> ( finite_card_list_nat(A3) = one_one_nat ) ) ).

% is_singleton_altdef
tff(fact_2804_is__singleton__altdef,axiom,
    ! [A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(is_singleton_set_nat,A3))
    <=> ( finite_card_set_nat(A3) = one_one_nat ) ) ).

% is_singleton_altdef
tff(fact_2805_is__singleton__altdef,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(is_singleton_nat,A3))
    <=> ( aa_set_nat_nat(finite_card_nat,A3) = one_one_nat ) ) ).

% is_singleton_altdef
tff(fact_2806_is__singleton__altdef,axiom,
    ! [A3: set_int] :
      ( pp(aa_set_int_bool(is_singleton_int,A3))
    <=> ( finite_card_int(A3) = one_one_nat ) ) ).

% is_singleton_altdef
tff(fact_2807_finite__list,axiom,
    ! [A3: set_VEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A3)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A3 ) ).

% finite_list
tff(fact_2808_finite__list,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A3 ) ).

% finite_list
tff(fact_2809_finite__list,axiom,
    ! [A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ? [Xs2: list_int] : set_int2(Xs2) = A3 ) ).

% finite_list
tff(fact_2810_finite__list,axiom,
    ! [A3: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A3 ) ).

% finite_list
tff(fact_2811_finite__list,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ? [Xs2: list_P6011104703257516679at_nat] : set_Pr5648618587558075414at_nat(Xs2) = A3 ) ).

% finite_list
tff(fact_2812_finite__list,axiom,
    ! [A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ? [Xs2: list_Extended_enat] : set_Extended_enat2(Xs2) = A3 ) ).

% finite_list
tff(fact_2813_subset__code_I1_J,axiom,
    ! [Xs: list_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Xs)),B2))
    <=> ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),set_real2(Xs)))
         => pp(aa_set_real_bool(member_real(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_2814_subset__code_I1_J,axiom,
    ! [Xs: list_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(set_bool2(Xs)),B2))
    <=> ! [X3: bool] :
          ( pp(aa_set_bool_bool(member_bool(X3),set_bool2(Xs)))
         => pp(aa_set_bool_bool(member_bool(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_2815_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),B2))
    <=> ! [X3: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(Xs)))
         => pp(aa_set_set_nat_bool(member_set_nat(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_2816_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B2)
    <=> ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
         => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),B2)) ) ) ).

% subset_code(1)
tff(fact_2817_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Xs)),B2))
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
         => pp(aa_set_nat_bool(member_nat(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_2818_subset__code_I1_J,axiom,
    ! [Xs: list_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),B2))
    <=> ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
         => pp(aa_set_int_bool(member_int(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_2819_case4_I8_J,axiom,
    ( ( mi = ma )
   => ! [X4: vEBT_VEBT] :
        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(treeList2)))
       => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_13)) ) ) ).

% case4(8)
tff(fact_2820_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
      <=> pp(Y) )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ( pp(Y)
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fFalse,fFalse))) ) )
         => ( ! [Uv2: bool] :
                ( ( X = vEBT_Leaf(fTrue,Uv2) )
               => ( ~ pp(Y)
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fTrue,Uv2))) ) )
           => ( ! [Uu2: bool] :
                  ( ( X = vEBT_Leaf(Uu2,fTrue) )
                 => ( ~ pp(Y)
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(Uu2,fTrue))) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
                   => ( pp(Y)
                     => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2))) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                     => ( ~ pp(Y)
                       => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_2821_case4_I13_J,axiom,
    vEBT_VEBT_set_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList2,summary2)) = vEBT_VEBT_set_vebt(sa) ).

% case4(13)
tff(fact_2822_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fFalse,fFalse))) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2))) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_2823_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
       => ( ! [Uv2: bool] :
              ( ( X = vEBT_Leaf(fTrue,Uv2) )
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fTrue,Uv2))) )
         => ( ! [Uu2: bool] :
                ( ( X = vEBT_Leaf(Uu2,fTrue) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(Uu2,fTrue))) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2))) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_2824_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
       => ( aa_int_int(aa_int_fun_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_2825_set__removeAll,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] : set_Pr5648618587558075414at_nat(remove3673390508374433037at_nat(X,Xs)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,set_Pr5648618587558075414at_nat(Xs)),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) ).

% set_removeAll
tff(fact_2826_set__removeAll,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] : set_VEBT_VEBT2(removeAll_VEBT_VEBT(X,Xs)) = minus_5127226145743854075T_VEBT(set_VEBT_VEBT2(Xs),insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)) ).

% set_removeAll
tff(fact_2827_set__removeAll,axiom,
    ! [X: real,Xs: list_real] : set_real2(removeAll_real(X,Xs)) = minus_minus_set_real(set_real2(Xs),insert_real(X,bot_bot_set_real)) ).

% set_removeAll
tff(fact_2828_set__removeAll,axiom,
    ! [X: bool,Xs: list_bool] : set_bool2(removeAll_bool(X,Xs)) = minus_minus_set_bool(set_bool2(Xs),insert_bool(X,bot_bot_set_bool)) ).

% set_removeAll
tff(fact_2829_set__removeAll,axiom,
    ! [X: int,Xs: list_int] : set_int2(removeAll_int(X,Xs)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,set_int2(Xs)),insert_int(X,bot_bot_set_int)) ).

% set_removeAll
tff(fact_2830_set__removeAll,axiom,
    ! [X: nat,Xs: list_nat] : set_nat2(removeAll_nat(X,Xs)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,set_nat2(Xs)),insert_nat(X,bot_bot_set_nat)) ).

% set_removeAll
tff(fact_2831_inthall,axiom,
    ! [Xs: list_real,P: fun_real_bool,N: nat] :
      ( ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),set_real2(Xs)))
         => pp(aa_real_bool(P,X5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
       => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),N))) ) ) ).

% inthall
tff(fact_2832_inthall,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_bool,N: nat] :
      ( ! [X5: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X5),set_set_nat2(Xs)))
         => pp(aa_set_nat_bool(P,X5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
       => pp(aa_set_nat_bool(P,nth_set_nat(Xs,N))) ) ) ).

% inthall
tff(fact_2833_inthall,axiom,
    ! [Xs: list_int,P: fun_int_bool,N: nat] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),set_int2(Xs)))
         => pp(aa_int_bool(P,X5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).

% inthall
tff(fact_2834_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,N: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,X5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,N))) ) ) ).

% inthall
tff(fact_2835_inthall,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,N: nat] :
      ( ! [X5: bool] :
          ( pp(aa_set_bool_bool(member_bool(X5),set_bool2(Xs)))
         => pp(aa_bool_bool(P,X5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% inthall
tff(fact_2836_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,N: nat] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% inthall
tff(fact_2837_nat__ivt__aux,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
         => ? [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_2838_case4_I9_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,mi),ma)) ).

% case4(9)
tff(fact_2839_case4_I3_J,axiom,
    pp(aa_nat_bool(vEBT_invar_vebt(summary2),m)) ).

% case4(3)
tff(fact_2840_abs__idempotent,axiom,
    ! [A2: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A2)) = aa_int_int(abs_abs_int,A2) ).

% abs_idempotent
tff(fact_2841_abs__idempotent,axiom,
    ! [A2: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A2)) = aa_real_real(abs_abs_real,A2) ).

% abs_idempotent
tff(fact_2842_abs__idempotent,axiom,
    ! [A2: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A2)) = aa_rat_rat(abs_abs_rat,A2) ).

% abs_idempotent
tff(fact_2843_abs__idempotent,axiom,
    ! [A2: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A2)) = abs_abs_Code_integer(A2) ).

% abs_idempotent
tff(fact_2844_case4_I2_J,axiom,
    ! [S2: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(S2),m))
     => ( ( vEBT_VEBT_set_vebt(summary2) = vEBT_VEBT_set_vebt(S2) )
       => ( S2 = summary2 ) ) ) ).

% case4(2)
tff(fact_2845_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_2846_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_2847_abs__0,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_2848_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_2849_abs__0__eq,axiom,
    ! [A2: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A2) )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_2850_abs__0__eq,axiom,
    ! [A2: real] :
      ( ( zero_zero_real = aa_real_real(abs_abs_real,A2) )
    <=> ( A2 = zero_zero_real ) ) ).

% abs_0_eq
tff(fact_2851_abs__0__eq,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat = aa_rat_rat(abs_abs_rat,A2) )
    <=> ( A2 = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_2852_abs__0__eq,axiom,
    ! [A2: int] :
      ( ( zero_zero_int = aa_int_int(abs_abs_int,A2) )
    <=> ( A2 = zero_zero_int ) ) ).

% abs_0_eq
tff(fact_2853_abs__eq__0,axiom,
    ! [A2: code_integer] :
      ( ( abs_abs_Code_integer(A2) = zero_z3403309356797280102nteger )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_2854_abs__eq__0,axiom,
    ! [A2: real] :
      ( ( aa_real_real(abs_abs_real,A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% abs_eq_0
tff(fact_2855_abs__eq__0,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_2856_abs__eq__0,axiom,
    ! [A2: int] :
      ( ( aa_int_int(abs_abs_int,A2) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% abs_eq_0
tff(fact_2857_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_2858_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_2859_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_2860_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_2861_abs__minus__cancel,axiom,
    ! [A2: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A2)) = aa_int_int(abs_abs_int,A2) ).

% abs_minus_cancel
tff(fact_2862_abs__minus__cancel,axiom,
    ! [A2: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A2)) = aa_real_real(abs_abs_real,A2) ).

% abs_minus_cancel
tff(fact_2863_abs__minus__cancel,axiom,
    ! [A2: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A2)) = aa_rat_rat(abs_abs_rat,A2) ).

% abs_minus_cancel
tff(fact_2864_abs__minus__cancel,axiom,
    ! [A2: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)) = abs_abs_Code_integer(A2) ).

% abs_minus_cancel
tff(fact_2865_abs__of__nat,axiom,
    ! [N: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ).

% abs_of_nat
tff(fact_2866_abs__of__nat,axiom,
    ! [N: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = aa_nat_Code_integer(semiri4939895301339042750nteger,N) ).

% abs_of_nat
tff(fact_2867_abs__of__nat,axiom,
    ! [N: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% abs_of_nat
tff(fact_2868_abs__of__nat,axiom,
    ! [N: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% abs_of_nat
tff(fact_2869_abs__le__zero__iff,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A2)),zero_z3403309356797280102nteger))
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_2870_abs__le__zero__iff,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),zero_zero_real))
    <=> ( A2 = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_2871_abs__le__zero__iff,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A2)),zero_zero_rat))
    <=> ( A2 = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_2872_abs__le__zero__iff,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),zero_zero_int))
    <=> ( A2 = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_2873_abs__le__self__iff,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A2)),A2))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)) ) ).

% abs_le_self_iff
tff(fact_2874_abs__le__self__iff,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),A2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2)) ) ).

% abs_le_self_iff
tff(fact_2875_abs__le__self__iff,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A2)),A2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2)) ) ).

% abs_le_self_iff
tff(fact_2876_abs__le__self__iff,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),A2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2)) ) ).

% abs_le_self_iff
tff(fact_2877_abs__of__nonneg,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => ( abs_abs_Code_integer(A2) = A2 ) ) ).

% abs_of_nonneg
tff(fact_2878_abs__of__nonneg,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( aa_real_real(abs_abs_real,A2) = A2 ) ) ).

% abs_of_nonneg
tff(fact_2879_abs__of__nonneg,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( aa_rat_rat(abs_abs_rat,A2) = A2 ) ) ).

% abs_of_nonneg
tff(fact_2880_abs__of__nonneg,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( aa_int_int(abs_abs_int,A2) = A2 ) ) ).

% abs_of_nonneg
tff(fact_2881_zero__less__abs__iff,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A2)))
    <=> ( A2 != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_2882_zero__less__abs__iff,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A2)))
    <=> ( A2 != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_2883_zero__less__abs__iff,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A2)))
    <=> ( A2 != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_2884_zero__less__abs__iff,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A2)))
    <=> ( A2 != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_2885_zero__le__divide__abs__iff,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(abs_abs_real,B3))))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
        | ( B3 = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_2886_zero__le__divide__abs__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),aa_rat_rat(abs_abs_rat,B3))))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
        | ( B3 = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_2887_divide__le__0__abs__iff,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(abs_abs_real,B3))),zero_zero_real))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
        | ( B3 = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_2888_divide__le__0__abs__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),aa_rat_rat(abs_abs_rat,B3))),zero_zero_rat))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
        | ( B3 = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_2889_abs__of__nonpos,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
     => ( aa_real_real(abs_abs_real,A2) = aa_real_real(uminus_uminus_real,A2) ) ) ).

% abs_of_nonpos
tff(fact_2890_abs__of__nonpos,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger))
     => ( abs_abs_Code_integer(A2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) ) ) ).

% abs_of_nonpos
tff(fact_2891_abs__of__nonpos,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
     => ( aa_rat_rat(abs_abs_rat,A2) = aa_rat_rat(uminus_uminus_rat,A2) ) ) ).

% abs_of_nonpos
tff(fact_2892_abs__of__nonpos,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( aa_int_int(abs_abs_int,A2) = aa_int_int(uminus_uminus_int,A2) ) ) ).

% abs_of_nonpos
tff(fact_2893_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int))
    <=> ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_2894_aa,axiom,
    pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,insert_nat(mi,insert_nat(ma,bot_bot_set_nat))),vEBT_VEBT_set_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList2,summary2)))) ).

% aa
tff(fact_2895_case4_I6_J,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% case4(6)
tff(fact_2896_abs__le__D1,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),B3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ).

% abs_le_D1
tff(fact_2897_abs__le__D1,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A2)),B3))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B3)) ) ).

% abs_le_D1
tff(fact_2898_abs__le__D1,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A2)),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ).

% abs_le_D1
tff(fact_2899_abs__le__D1,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ).

% abs_le_D1
tff(fact_2900_abs__ge__self,axiom,
    ! [A2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(abs_abs_real,A2))) ).

% abs_ge_self
tff(fact_2901_abs__ge__self,axiom,
    ! [A2: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),abs_abs_Code_integer(A2))) ).

% abs_ge_self
tff(fact_2902_abs__ge__self,axiom,
    ! [A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(abs_abs_rat,A2))) ).

% abs_ge_self
tff(fact_2903_abs__ge__self,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(abs_abs_int,A2))) ).

% abs_ge_self
tff(fact_2904_abs__eq__0__iff,axiom,
    ! [A2: code_integer] :
      ( ( abs_abs_Code_integer(A2) = zero_z3403309356797280102nteger )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_2905_abs__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( aa_real_real(abs_abs_real,A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% abs_eq_0_iff
tff(fact_2906_abs__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_2907_abs__eq__0__iff,axiom,
    ! [A2: int] :
      ( ( aa_int_int(abs_abs_int,A2) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% abs_eq_0_iff
tff(fact_2908_abs__minus__commute,axiom,
    ! [A2: code_integer,B3: code_integer] : abs_abs_Code_integer(minus_8373710615458151222nteger(A2,B3)) = abs_abs_Code_integer(minus_8373710615458151222nteger(B3,A2)) ).

% abs_minus_commute
tff(fact_2909_abs__minus__commute,axiom,
    ! [A2: real,B3: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),A2)) ).

% abs_minus_commute
tff(fact_2910_abs__minus__commute,axiom,
    ! [A2: rat,B3: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B3),A2)) ).

% abs_minus_commute
tff(fact_2911_abs__minus__commute,axiom,
    ! [A2: int,B3: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B3),A2)) ).

% abs_minus_commute
tff(fact_2912_infinite__int__iff__unbounded__le,axiom,
    ! [S: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S))
    <=> ! [M3: int] :
        ? [N3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M3),aa_int_int(abs_abs_int,N3)))
          & pp(aa_set_int_bool(member_int(N3),S)) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_2913_infinite__int__iff__unbounded,axiom,
    ! [S: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S))
    <=> ! [M3: int] :
        ? [N3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M3),aa_int_int(abs_abs_int,N3)))
          & pp(aa_set_int_bool(member_int(N3),S)) ) ) ).

% infinite_int_iff_unbounded
tff(fact_2914_abs__ge__zero,axiom,
    ! [A2: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A2))) ).

% abs_ge_zero
tff(fact_2915_abs__ge__zero,axiom,
    ! [A2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A2))) ).

% abs_ge_zero
tff(fact_2916_abs__ge__zero,axiom,
    ! [A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A2))) ).

% abs_ge_zero
tff(fact_2917_abs__ge__zero,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A2))) ).

% abs_ge_zero
tff(fact_2918_abs__of__pos,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
     => ( abs_abs_Code_integer(A2) = A2 ) ) ).

% abs_of_pos
tff(fact_2919_abs__of__pos,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( aa_real_real(abs_abs_real,A2) = A2 ) ) ).

% abs_of_pos
tff(fact_2920_abs__of__pos,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( aa_rat_rat(abs_abs_rat,A2) = A2 ) ) ).

% abs_of_pos
tff(fact_2921_abs__of__pos,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( aa_int_int(abs_abs_int,A2) = A2 ) ) ).

% abs_of_pos
tff(fact_2922_abs__not__less__zero,axiom,
    ! [A2: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A2)),zero_z3403309356797280102nteger)) ).

% abs_not_less_zero
tff(fact_2923_abs__not__less__zero,axiom,
    ! [A2: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A2)),zero_zero_real)) ).

% abs_not_less_zero
tff(fact_2924_abs__not__less__zero,axiom,
    ! [A2: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A2)),zero_zero_rat)) ).

% abs_not_less_zero
tff(fact_2925_abs__not__less__zero,axiom,
    ! [A2: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A2)),zero_zero_int)) ).

% abs_not_less_zero
tff(fact_2926_abs__triangle__ineq2__sym,axiom,
    ! [A2: code_integer,B3: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B3))),abs_abs_Code_integer(minus_8373710615458151222nteger(B3,A2)))) ).

% abs_triangle_ineq2_sym
tff(fact_2927_abs__triangle__ineq2__sym,axiom,
    ! [A2: real,B3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B3))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),A2)))) ).

% abs_triangle_ineq2_sym
tff(fact_2928_abs__triangle__ineq2__sym,axiom,
    ! [A2: rat,B3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B3))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B3),A2)))) ).

% abs_triangle_ineq2_sym
tff(fact_2929_abs__triangle__ineq2__sym,axiom,
    ! [A2: int,B3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B3))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B3),A2)))) ).

% abs_triangle_ineq2_sym
tff(fact_2930_abs__triangle__ineq3,axiom,
    ! [A2: code_integer,B3: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B3)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A2,B3)))) ).

% abs_triangle_ineq3
tff(fact_2931_abs__triangle__ineq3,axiom,
    ! [A2: real,B3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B3)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)))) ).

% abs_triangle_ineq3
tff(fact_2932_abs__triangle__ineq3,axiom,
    ! [A2: rat,B3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B3)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)))) ).

% abs_triangle_ineq3
tff(fact_2933_abs__triangle__ineq3,axiom,
    ! [A2: int,B3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B3)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)))) ).

% abs_triangle_ineq3
tff(fact_2934_abs__triangle__ineq2,axiom,
    ! [A2: code_integer,B3: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B3))),abs_abs_Code_integer(minus_8373710615458151222nteger(A2,B3)))) ).

% abs_triangle_ineq2
tff(fact_2935_abs__triangle__ineq2,axiom,
    ! [A2: real,B3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B3))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)))) ).

% abs_triangle_ineq2
tff(fact_2936_abs__triangle__ineq2,axiom,
    ! [A2: rat,B3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B3))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)))) ).

% abs_triangle_ineq2
tff(fact_2937_abs__triangle__ineq2,axiom,
    ! [A2: int,B3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B3))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)))) ).

% abs_triangle_ineq2
tff(fact_2938_nonzero__abs__divide,axiom,
    ! [B3: rat,A2: rat] :
      ( ( B3 != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B3)) ) ) ).

% nonzero_abs_divide
tff(fact_2939_nonzero__abs__divide,axiom,
    ! [B3: real,A2: real] :
      ( ( B3 != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B3)) ) ) ).

% nonzero_abs_divide
tff(fact_2940_abs__ge__minus__self,axiom,
    ! [A2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(abs_abs_real,A2))) ).

% abs_ge_minus_self
tff(fact_2941_abs__ge__minus__self,axiom,
    ! [A2: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),abs_abs_Code_integer(A2))) ).

% abs_ge_minus_self
tff(fact_2942_abs__ge__minus__self,axiom,
    ! [A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(abs_abs_rat,A2))) ).

% abs_ge_minus_self
tff(fact_2943_abs__ge__minus__self,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),aa_int_int(abs_abs_int,A2))) ).

% abs_ge_minus_self
tff(fact_2944_abs__le__iff,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),B3))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),B3)) ) ) ).

% abs_le_iff
tff(fact_2945_abs__le__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A2)),B3))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B3))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B3)) ) ) ).

% abs_le_iff
tff(fact_2946_abs__le__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A2)),B3))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),B3)) ) ) ).

% abs_le_iff
tff(fact_2947_abs__le__iff,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),B3))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),B3)) ) ) ).

% abs_le_iff
tff(fact_2948_abs__le__D2,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),B3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),B3)) ) ).

% abs_le_D2
tff(fact_2949_abs__le__D2,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A2)),B3))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B3)) ) ).

% abs_le_D2
tff(fact_2950_abs__le__D2,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A2)),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),B3)) ) ).

% abs_le_D2
tff(fact_2951_abs__le__D2,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),B3)) ) ).

% abs_le_D2
tff(fact_2952_abs__leI,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),B3)) ) ) ).

% abs_leI
tff(fact_2953_abs__leI,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B3))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B3))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A2)),B3)) ) ) ).

% abs_leI
tff(fact_2954_abs__leI,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A2)),B3)) ) ) ).

% abs_leI
tff(fact_2955_abs__leI,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),B3)) ) ) ).

% abs_leI
tff(fact_2956_abs__less__iff,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A2)),B3))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A2)),B3)) ) ) ).

% abs_less_iff
tff(fact_2957_abs__less__iff,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A2)),B3))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A2)),B3)) ) ) ).

% abs_less_iff
tff(fact_2958_abs__less__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A2)),B3))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A2)),B3)) ) ) ).

% abs_less_iff
tff(fact_2959_abs__less__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A2)),B3))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B3))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B3)) ) ) ).

% abs_less_iff
tff(fact_2960_nth__equalityI,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys2) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
           => ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys2),I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_2961_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys2) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
           => ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Ys2,I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_2962_nth__equalityI,axiom,
    ! [Xs: list_bool,Ys2: list_bool] :
      ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys2) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
           => ( pp(aa_nat_bool(nth_bool(Xs),I2))
            <=> pp(aa_nat_bool(nth_bool(Ys2),I2)) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_2963_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys2) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys2),I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_2964_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
         => ? [X_1: int] : pp(aa_int_bool(aa_nat_fun_int_bool(P,I4),X_1)) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
             => pp(aa_int_bool(aa_nat_fun_int_bool(P,I4),aa_nat_int(nth_int(Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_2965_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na2351039247863542899T_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
         => ? [X_1: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I4),X_1)) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
             => pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I4),nth_VEBT_VEBT(Xs3,I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_2966_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na9137082442855928016l_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
         => ? [X_1: bool] : pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I4),X_1)) )
    <=> ? [Xs3: list_bool] :
          ( ( size_size_list_bool(Xs3) = K )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
             => pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I4),aa_nat_bool(nth_bool(Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_2967_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
         => ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I4),X_1)) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I4),aa_nat_nat(nth_nat(Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_2968_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys2) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
           => ( aa_nat_int(nth_int(Xs),I4) = aa_nat_int(nth_int(Ys2),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_2969_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( Xs = Ys2 )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys2) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs)))
           => ( nth_VEBT_VEBT(Xs,I4) = nth_VEBT_VEBT(Ys2,I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_2970_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_bool,Ys2: list_bool] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys2) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Xs)))
           => ( pp(aa_nat_bool(nth_bool(Xs),I4))
            <=> pp(aa_nat_bool(nth_bool(Ys2),I4)) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_2971_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys2) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I4) = aa_nat_nat(nth_nat(Ys2),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_2972_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),E)) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_2973_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),E)) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_2974_eq__abs__iff_H,axiom,
    ! [A2: real,B3: real] :
      ( ( A2 = aa_real_real(abs_abs_real,B3) )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
        & ( ( B3 = A2 )
          | ( B3 = aa_real_real(uminus_uminus_real,A2) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2975_eq__abs__iff_H,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( A2 = abs_abs_Code_integer(B3) )
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
        & ( ( B3 = A2 )
          | ( B3 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2976_eq__abs__iff_H,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 = aa_rat_rat(abs_abs_rat,B3) )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
        & ( ( B3 = A2 )
          | ( B3 = aa_rat_rat(uminus_uminus_rat,A2) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2977_eq__abs__iff_H,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 = aa_int_int(abs_abs_int,B3) )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
        & ( ( B3 = A2 )
          | ( B3 = aa_int_int(uminus_uminus_int,A2) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2978_abs__eq__iff_H,axiom,
    ! [A2: real,B3: real] :
      ( ( aa_real_real(abs_abs_real,A2) = B3 )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3))
        & ( ( A2 = B3 )
          | ( A2 = aa_real_real(uminus_uminus_real,B3) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2979_abs__eq__iff_H,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( abs_abs_Code_integer(A2) = B3 )
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B3))
        & ( ( A2 = B3 )
          | ( A2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2980_abs__eq__iff_H,axiom,
    ! [A2: rat,B3: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A2) = B3 )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3))
        & ( ( A2 = B3 )
          | ( A2 = aa_rat_rat(uminus_uminus_rat,B3) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2981_abs__eq__iff_H,axiom,
    ! [A2: int,B3: int] :
      ( ( aa_int_int(abs_abs_int,A2) = B3 )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3))
        & ( ( A2 = B3 )
          | ( A2 = aa_int_int(uminus_uminus_int,B3) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2982_abs__minus__le__zero,axiom,
    ! [A2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,A2))),zero_zero_real)) ).

% abs_minus_le_zero
tff(fact_2983_abs__minus__le__zero,axiom,
    ! [A2: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,abs_abs_Code_integer(A2))),zero_z3403309356797280102nteger)) ).

% abs_minus_le_zero
tff(fact_2984_abs__minus__le__zero,axiom,
    ! [A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(abs_abs_rat,A2))),zero_zero_rat)) ).

% abs_minus_le_zero
tff(fact_2985_abs__minus__le__zero,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A2))),zero_zero_int)) ).

% abs_minus_le_zero
tff(fact_2986_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(abs_abs_rat,X)),Y) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% abs_div_pos
tff(fact_2987_abs__div__pos,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,X)),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% abs_div_pos
tff(fact_2988_abs__if,axiom,
    ! [A2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
       => ( aa_int_int(abs_abs_int,A2) = aa_int_int(uminus_uminus_int,A2) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
       => ( aa_int_int(abs_abs_int,A2) = A2 ) ) ) ).

% abs_if
tff(fact_2989_abs__if,axiom,
    ! [A2: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A2) = aa_real_real(uminus_uminus_real,A2) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A2) = A2 ) ) ) ).

% abs_if
tff(fact_2990_abs__if,axiom,
    ! [A2: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A2) = aa_rat_rat(uminus_uminus_rat,A2) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A2) = A2 ) ) ) ).

% abs_if
tff(fact_2991_abs__if,axiom,
    ! [A2: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(A2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(A2) = A2 ) ) ) ).

% abs_if
tff(fact_2992_abs__of__neg,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( aa_int_int(abs_abs_int,A2) = aa_int_int(uminus_uminus_int,A2) ) ) ).

% abs_of_neg
tff(fact_2993_abs__of__neg,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( aa_real_real(abs_abs_real,A2) = aa_real_real(uminus_uminus_real,A2) ) ) ).

% abs_of_neg
tff(fact_2994_abs__of__neg,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( aa_rat_rat(abs_abs_rat,A2) = aa_rat_rat(uminus_uminus_rat,A2) ) ) ).

% abs_of_neg
tff(fact_2995_abs__of__neg,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger))
     => ( abs_abs_Code_integer(A2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) ) ) ).

% abs_of_neg
tff(fact_2996_abs__if__raw,axiom,
    ! [X4: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X4) = aa_int_int(uminus_uminus_int,X4) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X4) = X4 ) ) ) ).

% abs_if_raw
tff(fact_2997_abs__if__raw,axiom,
    ! [X4: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X4) = aa_real_real(uminus_uminus_real,X4) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X4) = X4 ) ) ) ).

% abs_if_raw
tff(fact_2998_abs__if__raw,axiom,
    ! [X4: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,X4) = aa_rat_rat(uminus_uminus_rat,X4) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,X4) = X4 ) ) ) ).

% abs_if_raw
tff(fact_2999_abs__if__raw,axiom,
    ! [X4: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X4),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X4) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X4) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X4),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X4) = X4 ) ) ) ).

% abs_if_raw
tff(fact_3000_length__removeAll__less__eq,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(removeAll_VEBT_VEBT(X,Xs))),size_s6755466524823107622T_VEBT(Xs))) ).

% length_removeAll_less_eq
tff(fact_3001_length__removeAll__less__eq,axiom,
    ! [X: bool,Xs: list_bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(removeAll_bool(X,Xs))),size_size_list_bool(Xs))) ).

% length_removeAll_less_eq
tff(fact_3002_length__removeAll__less__eq,axiom,
    ! [X: nat,Xs: list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(removeAll_nat(X,Xs))),size_size_list_nat(Xs))) ).

% length_removeAll_less_eq
tff(fact_3003_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_bool] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
         => pp(aa_int_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_3004_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_3005_all__set__conv__all__nth,axiom,
    ! [Xs: list_bool,P: fun_bool_bool] :
      ( ! [X3: bool] :
          ( pp(aa_set_bool_bool(member_bool(X3),set_bool2(Xs)))
         => pp(aa_bool_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Xs)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_3006_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_bool] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_3007_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_bool,X: real] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
         => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),I2))) )
     => ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
       => pp(aa_real_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_3008_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_bool,X: set_nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s3254054031482475050et_nat(Xs)))
         => pp(aa_set_nat_bool(P,nth_set_nat(Xs,I2))) )
     => ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
       => pp(aa_set_nat_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_3009_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_bool,X: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I2))) )
     => ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
       => pp(aa_int_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_3010_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I2))) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
       => pp(aa_VEBT_VEBT_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_3011_all__nth__imp__all__set,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,X: bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I2))) )
     => ( pp(aa_set_bool_bool(member_bool(X),set_bool2(Xs)))
       => pp(aa_bool_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_3012_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,X: nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I2))) )
     => ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
       => pp(aa_nat_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_3013_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_real(Xs)))
          & ( aa_nat_real(nth_real(Xs),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3014_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s3254054031482475050et_nat(Xs)))
          & ( nth_set_nat(Xs,I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3015_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
          & ( aa_nat_int(nth_int(Xs),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3016_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs)))
          & ( nth_VEBT_VEBT(Xs,I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3017_in__set__conv__nth,axiom,
    ! [X: bool,Xs: list_bool] :
      ( pp(aa_set_bool_bool(member_bool(X),set_bool2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Xs)))
          & ( pp(aa_nat_bool(nth_bool(Xs),I4))
          <=> pp(X) ) ) ) ).

% in_set_conv_nth
tff(fact_3018_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
          & ( aa_nat_nat(nth_nat(Xs),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3019_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P: fun_int_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => ( ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),set_int2(Xs)))
           => pp(aa_int_bool(P,X5)) )
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_3020_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( ! [X5: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(Xs)))
           => pp(aa_VEBT_VEBT_bool(P,X5)) )
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,N))) ) ) ).

% list_ball_nth
tff(fact_3021_list__ball__nth,axiom,
    ! [N: nat,Xs: list_bool,P: fun_bool_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => ( ! [X5: bool] :
            ( pp(aa_set_bool_bool(member_bool(X5),set_bool2(Xs)))
           => pp(aa_bool_bool(P,X5)) )
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_3022_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => ( ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(Xs)))
           => pp(aa_nat_bool(P,X5)) )
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_3023_nth__mem,axiom,
    ! [N: nat,Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => pp(aa_set_real_bool(member_real(aa_nat_real(nth_real(Xs),N)),set_real2(Xs))) ) ).

% nth_mem
tff(fact_3024_nth__mem,axiom,
    ! [N: nat,Xs: list_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
     => pp(aa_set_set_nat_bool(member_set_nat(nth_set_nat(Xs,N)),set_set_nat2(Xs))) ) ).

% nth_mem
tff(fact_3025_nth__mem,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => pp(aa_set_int_bool(member_int(aa_nat_int(nth_int(Xs),N)),set_int2(Xs))) ) ).

% nth_mem
tff(fact_3026_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,nth_VEBT_VEBT(Xs,N)),set_VEBT_VEBT2(Xs))) ) ).

% nth_mem
tff(fact_3027_nth__mem,axiom,
    ! [N: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => pp(aa_set_bool_bool(member_bool(aa_nat_bool(nth_bool(Xs),N)),set_bool2(Xs))) ) ).

% nth_mem
tff(fact_3028_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => pp(aa_set_nat_bool(member_nat(aa_nat_nat(nth_nat(Xs),N)),set_nat2(Xs))) ) ).

% nth_mem
tff(fact_3029_zabs__def,axiom,
    ! [I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I) = aa_int_int(uminus_uminus_int,I) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I) = I ) ) ) ).

% zabs_def
tff(fact_3030_length__removeAll__less,axiom,
    ! [X: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_real(removeAll_real(X,Xs))),size_size_list_real(Xs))) ) ).

% length_removeAll_less
tff(fact_3031_length__removeAll__less,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s3254054031482475050et_nat(removeAll_set_nat(X,Xs))),size_s3254054031482475050et_nat(Xs))) ) ).

% length_removeAll_less
tff(fact_3032_length__removeAll__less,axiom,
    ! [X: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(removeAll_int(X,Xs))),size_size_list_int(Xs))) ) ).

% length_removeAll_less
tff(fact_3033_length__removeAll__less,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(removeAll_VEBT_VEBT(X,Xs))),size_s6755466524823107622T_VEBT(Xs))) ) ).

% length_removeAll_less
tff(fact_3034_length__removeAll__less,axiom,
    ! [X: bool,Xs: list_bool] :
      ( pp(aa_set_bool_bool(member_bool(X),set_bool2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(removeAll_bool(X,Xs))),size_size_list_bool(Xs))) ) ).

% length_removeAll_less
tff(fact_3035_length__removeAll__less,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(removeAll_nat(X,Xs))),size_size_list_nat(Xs))) ) ).

% length_removeAll_less
tff(fact_3036_power__diff__power__eq,axiom,
    ! [A2: int,N: nat,M: nat] :
      ( ( A2 != zero_zero_int )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A2),M)),aa_nat_int(power_power_int(A2),N)) = aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A2),M)),aa_nat_int(power_power_int(A2),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_3037_power__diff__power__eq,axiom,
    ! [A2: nat,N: nat,M: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_3038_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,M)),K))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
           => ? [I2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I2))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
                & ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_3039_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3040_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3041_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3042_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3043_power__decreasing__iff,axiom,
    ! [B3: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_3044_power__decreasing__iff,axiom,
    ! [B3: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),one_one_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B3),M)),aa_nat_rat(power_power_rat(B3),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_3045_power__decreasing__iff,axiom,
    ! [B3: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_3046_power__decreasing__iff,axiom,
    ! [B3: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B3),M)),aa_nat_int(power_power_int(B3),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_3047_zero__less__power__abs__iff,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A2)),N)))
    <=> ( ( A2 != zero_z3403309356797280102nteger )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_3048_zero__less__power__abs__iff,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),N)))
    <=> ( ( A2 != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_3049_zero__less__power__abs__iff,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A2)),N)))
    <=> ( ( A2 != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_3050_zero__less__power__abs__iff,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A2)),N)))
    <=> ( ( A2 != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_3051_power__mono__iff,axiom,
    ! [A2: real,B3: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),N)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ) ) ) ).

% power_mono_iff
tff(fact_3052_power__mono__iff,axiom,
    ! [A2: rat,B3: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B3),N)))
          <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ) ) ) ).

% power_mono_iff
tff(fact_3053_power__mono__iff,axiom,
    ! [A2: nat,B3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),N)))
          <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3)) ) ) ) ) ).

% power_mono_iff
tff(fact_3054_power__mono__iff,axiom,
    ! [A2: int,B3: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(B3),N)))
          <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ) ) ) ).

% power_mono_iff
tff(fact_3055_power__increasing__iff,axiom,
    ! [B3: real,X: nat,Y: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_3056_power__increasing__iff,axiom,
    ! [B3: rat,X: nat,Y: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B3),X)),aa_nat_rat(power_power_rat(B3),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_3057_power__increasing__iff,axiom,
    ! [B3: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_3058_power__increasing__iff,axiom,
    ! [B3: int,X: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B3),X)),aa_nat_int(power_power_int(B3),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_3059_power__strict__decreasing__iff,axiom,
    ! [B3: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3060_power__strict__decreasing__iff,axiom,
    ! [B3: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),one_one_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B3),M)),aa_nat_rat(power_power_rat(B3),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3061_power__strict__decreasing__iff,axiom,
    ! [B3: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3062_power__strict__decreasing__iff,axiom,
    ! [B3: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B3),M)),aa_nat_int(power_power_int(B3),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3063_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B3: nat,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B3)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3064_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B3: nat,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B3)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3065_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B3: nat,W2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B3)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3066_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B3: nat,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B3)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3067_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W2: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B3)),W2)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3068_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W2: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B3)),W2)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3069_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B3)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3070_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W2: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B3)),W2)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3071_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B3: nat,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B3)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3072_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B3: nat,W2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B3)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3073_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B3: nat,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B3)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3074_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B3: nat,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B3)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3075_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N2: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2)
     => ~ ! [N2: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(suc,N2)) ) ).

% even_odd_cases
tff(fact_3076_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_3077_power__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(X)),some_nat(Y)) = some_nat(Z) ) ) ).

% power_shift
tff(fact_3078_add__left__cancel,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C) )
    <=> ( B3 = C ) ) ).

% add_left_cancel
tff(fact_3079_add__left__cancel,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C) )
    <=> ( B3 = C ) ) ).

% add_left_cancel
tff(fact_3080_add__left__cancel,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C) )
    <=> ( B3 = C ) ) ).

% add_left_cancel
tff(fact_3081_add__left__cancel,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C) )
    <=> ( B3 = C ) ) ).

% add_left_cancel
tff(fact_3082_add__right__cancel,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A2) )
    <=> ( B3 = C ) ) ).

% add_right_cancel
tff(fact_3083_add__right__cancel,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2) )
    <=> ( B3 = C ) ) ).

% add_right_cancel
tff(fact_3084_add__right__cancel,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2) )
    <=> ( B3 = C ) ) ).

% add_right_cancel
tff(fact_3085_add__right__cancel,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2) )
    <=> ( B3 = C ) ) ).

% add_right_cancel
tff(fact_3086_add__le__cancel__right,axiom,
    ! [A2: real,C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),C)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ).

% add_le_cancel_right
tff(fact_3087_add__le__cancel__right,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),C)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ).

% add_le_cancel_right
tff(fact_3088_add__le__cancel__right,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),C)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3)) ) ).

% add_le_cancel_right
tff(fact_3089_add__le__cancel__right,axiom,
    ! [A2: int,C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),C)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ).

% add_le_cancel_right
tff(fact_3090_add__le__cancel__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ).

% add_le_cancel_left
tff(fact_3091_add__le__cancel__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B3)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ).

% add_le_cancel_left
tff(fact_3092_add__le__cancel__left,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3)) ) ).

% add_le_cancel_left
tff(fact_3093_add__le__cancel__left,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ).

% add_le_cancel_left
tff(fact_3094_double__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_3095_double__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_3096_double__eq__0__iff,axiom,
    ! [A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_3097_add__0,axiom,
    ! [A2: literal] : plus_plus_literal(zero_zero_literal,A2) = A2 ).

% add_0
tff(fact_3098_add__0,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A2) = A2 ).

% add_0
tff(fact_3099_add__0,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A2) = A2 ).

% add_0
tff(fact_3100_add__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A2) = A2 ).

% add_0
tff(fact_3101_add__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A2) = A2 ).

% add_0
tff(fact_3102_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_3103_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_3104_add__cancel__right__right,axiom,
    ! [A2: real,B3: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3) )
    <=> ( B3 = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_3105_add__cancel__right__right,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3) )
    <=> ( B3 = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_3106_add__cancel__right__right,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3) )
    <=> ( B3 = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_3107_add__cancel__right__right,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3) )
    <=> ( B3 = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_3108_add__cancel__right__left,axiom,
    ! [A2: real,B3: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),A2) )
    <=> ( B3 = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_3109_add__cancel__right__left,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),A2) )
    <=> ( B3 = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_3110_add__cancel__right__left,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),A2) )
    <=> ( B3 = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_3111_add__cancel__right__left,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),A2) )
    <=> ( B3 = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_3112_add__cancel__left__right,axiom,
    ! [A2: real,B3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3) = A2 )
    <=> ( B3 = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_3113_add__cancel__left__right,axiom,
    ! [A2: rat,B3: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3) = A2 )
    <=> ( B3 = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_3114_add__cancel__left__right,axiom,
    ! [A2: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3) = A2 )
    <=> ( B3 = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_3115_add__cancel__left__right,axiom,
    ! [A2: int,B3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3) = A2 )
    <=> ( B3 = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_3116_add__cancel__left__left,axiom,
    ! [B3: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),A2) = A2 )
    <=> ( B3 = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_3117_add__cancel__left__left,axiom,
    ! [B3: rat,A2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),A2) = A2 )
    <=> ( B3 = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_3118_add__cancel__left__left,axiom,
    ! [B3: nat,A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),A2) = A2 )
    <=> ( B3 = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_3119_add__cancel__left__left,axiom,
    ! [B3: int,A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),A2) = A2 )
    <=> ( B3 = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_3120_double__zero__sym,axiom,
    ! [A2: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2) )
    <=> ( A2 = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_3121_double__zero__sym,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),A2) )
    <=> ( A2 = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_3122_double__zero__sym,axiom,
    ! [A2: int] :
      ( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2) )
    <=> ( A2 = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_3123_add_Oright__neutral,axiom,
    ! [A2: literal] : plus_plus_literal(A2,zero_zero_literal) = A2 ).

% add.right_neutral
tff(fact_3124_add_Oright__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),zero_zero_real) = A2 ).

% add.right_neutral
tff(fact_3125_add_Oright__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),zero_zero_rat) = A2 ).

% add.right_neutral
tff(fact_3126_add_Oright__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),zero_zero_nat) = A2 ).

% add.right_neutral
tff(fact_3127_add_Oright__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),zero_zero_int) = A2 ).

% add.right_neutral
tff(fact_3128_add__less__cancel__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) ) ).

% add_less_cancel_left
tff(fact_3129_add__less__cancel__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B3)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) ) ).

% add_less_cancel_left
tff(fact_3130_add__less__cancel__left,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3)) ) ).

% add_less_cancel_left
tff(fact_3131_add__less__cancel__left,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3)) ) ).

% add_less_cancel_left
tff(fact_3132_add__less__cancel__right,axiom,
    ! [A2: real,C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),C)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) ) ).

% add_less_cancel_right
tff(fact_3133_add__less__cancel__right,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),C)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) ) ).

% add_less_cancel_right
tff(fact_3134_add__less__cancel__right,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),C)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3)) ) ).

% add_less_cancel_right
tff(fact_3135_add__less__cancel__right,axiom,
    ! [A2: int,C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),C)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3)) ) ).

% add_less_cancel_right
tff(fact_3136_add__diff__cancel,axiom,
    ! [A2: real,B3: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),B3) = A2 ).

% add_diff_cancel
tff(fact_3137_add__diff__cancel,axiom,
    ! [A2: rat,B3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)),B3) = A2 ).

% add_diff_cancel
tff(fact_3138_add__diff__cancel,axiom,
    ! [A2: int,B3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)),B3) = A2 ).

% add_diff_cancel
tff(fact_3139_diff__add__cancel,axiom,
    ! [A2: real,B3: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)),B3) = A2 ).

% diff_add_cancel
tff(fact_3140_diff__add__cancel,axiom,
    ! [A2: rat,B3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)),B3) = A2 ).

% diff_add_cancel
tff(fact_3141_diff__add__cancel,axiom,
    ! [A2: int,B3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)),B3) = A2 ).

% diff_add_cancel
tff(fact_3142_add__diff__cancel__left,axiom,
    ! [C: real,A2: real,B3: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B3)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3) ).

% add_diff_cancel_left
tff(fact_3143_add__diff__cancel__left,axiom,
    ! [C: rat,A2: rat,B3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3) ).

% add_diff_cancel_left
tff(fact_3144_add__diff__cancel__left,axiom,
    ! [C: nat,A2: nat,B3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B3) ).

% add_diff_cancel_left
tff(fact_3145_add__diff__cancel__left,axiom,
    ! [C: int,A2: int,B3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B3)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3) ).

% add_diff_cancel_left
tff(fact_3146_add__diff__cancel__left_H,axiom,
    ! [A2: real,B3: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),A2) = B3 ).

% add_diff_cancel_left'
tff(fact_3147_add__diff__cancel__left_H,axiom,
    ! [A2: rat,B3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)),A2) = B3 ).

% add_diff_cancel_left'
tff(fact_3148_add__diff__cancel__left_H,axiom,
    ! [A2: nat,B3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3)),A2) = B3 ).

% add_diff_cancel_left'
tff(fact_3149_add__diff__cancel__left_H,axiom,
    ! [A2: int,B3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)),A2) = B3 ).

% add_diff_cancel_left'
tff(fact_3150_add__diff__cancel__right,axiom,
    ! [A2: real,C: real,B3: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3) ).

% add_diff_cancel_right
tff(fact_3151_add__diff__cancel__right,axiom,
    ! [A2: rat,C: rat,B3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3) ).

% add_diff_cancel_right
tff(fact_3152_add__diff__cancel__right,axiom,
    ! [A2: nat,C: nat,B3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B3) ).

% add_diff_cancel_right
tff(fact_3153_add__diff__cancel__right,axiom,
    ! [A2: int,C: int,B3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3) ).

% add_diff_cancel_right
tff(fact_3154_add__diff__cancel__right_H,axiom,
    ! [A2: real,B3: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),B3) = A2 ).

% add_diff_cancel_right'
tff(fact_3155_add__diff__cancel__right_H,axiom,
    ! [A2: rat,B3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)),B3) = A2 ).

% add_diff_cancel_right'
tff(fact_3156_add__diff__cancel__right_H,axiom,
    ! [A2: nat,B3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3)),B3) = A2 ).

% add_diff_cancel_right'
tff(fact_3157_add__diff__cancel__right_H,axiom,
    ! [A2: int,B3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)),B3) = A2 ).

% add_diff_cancel_right'
tff(fact_3158_add__minus__cancel,axiom,
    ! [A2: int,B3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A2)),B3)) = B3 ).

% add_minus_cancel
tff(fact_3159_add__minus__cancel,axiom,
    ! [A2: real,B3: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),B3)) = B3 ).

% add_minus_cancel
tff(fact_3160_add__minus__cancel,axiom,
    ! [A2: rat,B3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A2)),B3)) = B3 ).

% add_minus_cancel
tff(fact_3161_add__minus__cancel,axiom,
    ! [A2: code_integer,B3: code_integer] : plus_p5714425477246183910nteger(A2,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2),B3)) = B3 ).

% add_minus_cancel
tff(fact_3162_add__minus__cancel,axiom,
    ! [A2: complex,B3: complex] : aa_complex_complex(plus_plus_complex(A2),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),B3)) = B3 ).

% add_minus_cancel
tff(fact_3163_minus__add__cancel,axiom,
    ! [A2: int,B3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)) = B3 ).

% minus_add_cancel
tff(fact_3164_minus__add__cancel,axiom,
    ! [A2: real,B3: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)) = B3 ).

% minus_add_cancel
tff(fact_3165_minus__add__cancel,axiom,
    ! [A2: rat,B3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)) = B3 ).

% minus_add_cancel
tff(fact_3166_minus__add__cancel,axiom,
    ! [A2: code_integer,B3: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2),plus_p5714425477246183910nteger(A2,B3)) = B3 ).

% minus_add_cancel
tff(fact_3167_minus__add__cancel,axiom,
    ! [A2: complex,B3: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(plus_plus_complex(A2),B3)) = B3 ).

% minus_add_cancel
tff(fact_3168_minus__add__distrib,axiom,
    ! [A2: int,B3: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A2)),aa_int_int(uminus_uminus_int,B3)) ).

% minus_add_distrib
tff(fact_3169_minus__add__distrib,axiom,
    ! [A2: real,B3: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(uminus_uminus_real,B3)) ).

% minus_add_distrib
tff(fact_3170_minus__add__distrib,axiom,
    ! [A2: rat,B3: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(uminus_uminus_rat,B3)) ).

% minus_add_distrib
tff(fact_3171_minus__add__distrib,axiom,
    ! [A2: code_integer,B3: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A2,B3)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3)) ).

% minus_add_distrib
tff(fact_3172_minus__add__distrib,axiom,
    ! [A2: complex,B3: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A2),B3)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(uminus1482373934393186551omplex,B3)) ).

% minus_add_distrib
tff(fact_3173_abs__add__abs,axiom,
    ! [A2: code_integer,B3: code_integer] : abs_abs_Code_integer(plus_p5714425477246183910nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B3))) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B3)) ).

% abs_add_abs
tff(fact_3174_abs__add__abs,axiom,
    ! [A2: real,B3: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B3))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B3)) ).

% abs_add_abs
tff(fact_3175_abs__add__abs,axiom,
    ! [A2: rat,B3: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B3))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B3)) ).

% abs_add_abs
tff(fact_3176_abs__add__abs,axiom,
    ! [A2: int,B3: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B3)) ).

% abs_add_abs
tff(fact_3177_add__Suc__right,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% add_Suc_right
tff(fact_3178_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_3179_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_3180_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% nat_add_left_cancel_less
tff(fact_3181_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% nat_add_left_cancel_le
tff(fact_3182_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_3183_add__le__same__cancel1,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),A2)),B3))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) ) ).

% add_le_same_cancel1
tff(fact_3184_add__le__same__cancel1,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),A2)),B3))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) ) ).

% add_le_same_cancel1
tff(fact_3185_add__le__same__cancel1,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),A2)),B3))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat)) ) ).

% add_le_same_cancel1
tff(fact_3186_add__le__same__cancel1,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),A2)),B3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) ) ).

% add_le_same_cancel1
tff(fact_3187_add__le__same__cancel2,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),B3))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) ) ).

% add_le_same_cancel2
tff(fact_3188_add__le__same__cancel2,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)),B3))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) ) ).

% add_le_same_cancel2
tff(fact_3189_add__le__same__cancel2,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3)),B3))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat)) ) ).

% add_le_same_cancel2
tff(fact_3190_add__le__same__cancel2,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)),B3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) ) ).

% add_le_same_cancel2
tff(fact_3191_le__add__same__cancel1,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3)) ) ).

% le_add_same_cancel1
tff(fact_3192_le__add__same__cancel1,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3)) ) ).

% le_add_same_cancel1
tff(fact_3193_le__add__same__cancel1,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B3)) ) ).

% le_add_same_cancel1
tff(fact_3194_le__add__same__cancel1,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3)) ) ).

% le_add_same_cancel1
tff(fact_3195_le__add__same__cancel2,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3)) ) ).

% le_add_same_cancel2
tff(fact_3196_le__add__same__cancel2,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3)) ) ).

% le_add_same_cancel2
tff(fact_3197_le__add__same__cancel2,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),A2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B3)) ) ).

% le_add_same_cancel2
tff(fact_3198_le__add__same__cancel2,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3)) ) ).

% le_add_same_cancel2
tff(fact_3199_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_3200_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),A2)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_3201_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_3202_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_3203_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_3204_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_3205_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_3206_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_3207_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_3208_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_3209_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),A2)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_3210_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_3211_less__add__same__cancel2,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3)) ) ).

% less_add_same_cancel2
tff(fact_3212_less__add__same__cancel2,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3)) ) ).

% less_add_same_cancel2
tff(fact_3213_less__add__same__cancel2,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),A2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3)) ) ).

% less_add_same_cancel2
tff(fact_3214_less__add__same__cancel2,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3)) ) ).

% less_add_same_cancel2
tff(fact_3215_less__add__same__cancel1,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3)) ) ).

% less_add_same_cancel1
tff(fact_3216_less__add__same__cancel1,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3)) ) ).

% less_add_same_cancel1
tff(fact_3217_less__add__same__cancel1,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3)) ) ).

% less_add_same_cancel1
tff(fact_3218_less__add__same__cancel1,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3)) ) ).

% less_add_same_cancel1
tff(fact_3219_add__less__same__cancel2,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),B3))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ).

% add_less_same_cancel2
tff(fact_3220_add__less__same__cancel2,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)),B3))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ).

% add_less_same_cancel2
tff(fact_3221_add__less__same__cancel2,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3)),B3))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat)) ) ).

% add_less_same_cancel2
tff(fact_3222_add__less__same__cancel2,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)),B3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ).

% add_less_same_cancel2
tff(fact_3223_add__less__same__cancel1,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),A2)),B3))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ).

% add_less_same_cancel1
tff(fact_3224_add__less__same__cancel1,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),A2)),B3))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ).

% add_less_same_cancel1
tff(fact_3225_add__less__same__cancel1,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),A2)),B3))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat)) ) ).

% add_less_same_cancel1
tff(fact_3226_add__less__same__cancel1,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),A2)),B3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ).

% add_less_same_cancel1
tff(fact_3227_le__add__diff__inverse2,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)),B3) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_3228_le__add__diff__inverse2,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)),B3) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_3229_le__add__diff__inverse2,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B3)),B3) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_3230_le__add__diff__inverse2,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)),B3) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_3231_le__add__diff__inverse,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_3232_le__add__diff__inverse,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_3233_le__add__diff__inverse,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B3)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_3234_le__add__diff__inverse,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_3235_diff__add__zero,axiom,
    ! [A2: nat,B3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3)) = zero_zero_nat ).

% diff_add_zero
tff(fact_3236_add_Oright__inverse,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(uminus_uminus_int,A2)) = zero_zero_int ).

% add.right_inverse
tff(fact_3237_add_Oright__inverse,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(uminus_uminus_real,A2)) = zero_zero_real ).

% add.right_inverse
tff(fact_3238_add_Oright__inverse,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(uminus_uminus_rat,A2)) = zero_zero_rat ).

% add.right_inverse
tff(fact_3239_add_Oright__inverse,axiom,
    ! [A2: code_integer] : plus_p5714425477246183910nteger(A2,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_3240_add_Oright__inverse,axiom,
    ! [A2: complex] : aa_complex_complex(plus_plus_complex(A2),aa_complex_complex(uminus1482373934393186551omplex,A2)) = zero_zero_complex ).

% add.right_inverse
tff(fact_3241_ab__left__minus,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A2)),A2) = zero_zero_int ).

% ab_left_minus
tff(fact_3242_ab__left__minus,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),A2) = zero_zero_real ).

% ab_left_minus
tff(fact_3243_ab__left__minus,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A2)),A2) = zero_zero_rat ).

% ab_left_minus
tff(fact_3244_ab__left__minus,axiom,
    ! [A2: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2),A2) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_3245_ab__left__minus,axiom,
    ! [A2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),A2) = zero_zero_complex ).

% ab_left_minus
tff(fact_3246_power__inject__exp,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_3247_power__inject__exp,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2))
     => ( ( aa_nat_rat(power_power_rat(A2),M) = aa_nat_rat(power_power_rat(A2),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_3248_power__inject__exp,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A2))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_3249_power__inject__exp,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
     => ( ( aa_nat_int(power_power_int(A2),M) = aa_nat_int(power_power_int(A2),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_3250_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,N)) = zero_zero_rat ).

% power_0_Suc
tff(fact_3251_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,N)) = zero_zero_int ).

% power_0_Suc
tff(fact_3252_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,N)) = zero_zero_nat ).

% power_0_Suc
tff(fact_3253_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,N)) = zero_zero_real ).

% power_0_Suc
tff(fact_3254_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,N)) = zero_zero_complex ).

% power_0_Suc
tff(fact_3255_diff__minus__eq__add,axiom,
    ! [A2: int,B3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(uminus_uminus_int,B3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3) ).

% diff_minus_eq_add
tff(fact_3256_diff__minus__eq__add,axiom,
    ! [A2: real,B3: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(uminus_uminus_real,B3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3) ).

% diff_minus_eq_add
tff(fact_3257_diff__minus__eq__add,axiom,
    ! [A2: rat,B3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),aa_rat_rat(uminus_uminus_rat,B3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3) ).

% diff_minus_eq_add
tff(fact_3258_diff__minus__eq__add,axiom,
    ! [A2: code_integer,B3: code_integer] : minus_8373710615458151222nteger(A2,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3)) = plus_p5714425477246183910nteger(A2,B3) ).

% diff_minus_eq_add
tff(fact_3259_diff__minus__eq__add,axiom,
    ! [A2: complex,B3: complex] : aa_complex_complex(minus_minus_complex(A2),aa_complex_complex(uminus1482373934393186551omplex,B3)) = aa_complex_complex(plus_plus_complex(A2),B3) ).

% diff_minus_eq_add
tff(fact_3260_uminus__add__conv__diff,axiom,
    ! [A2: int,B3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A2)),B3) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B3),A2) ).

% uminus_add_conv_diff
tff(fact_3261_uminus__add__conv__diff,axiom,
    ! [A2: real,B3: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),B3) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),A2) ).

% uminus_add_conv_diff
tff(fact_3262_uminus__add__conv__diff,axiom,
    ! [A2: rat,B3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A2)),B3) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B3),A2) ).

% uminus_add_conv_diff
tff(fact_3263_uminus__add__conv__diff,axiom,
    ! [A2: code_integer,B3: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2),B3) = minus_8373710615458151222nteger(B3,A2) ).

% uminus_add_conv_diff
tff(fact_3264_uminus__add__conv__diff,axiom,
    ! [A2: complex,B3: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),B3) = aa_complex_complex(minus_minus_complex(B3),A2) ).

% uminus_add_conv_diff
tff(fact_3265_power__Suc0__right,axiom,
    ! [A2: int] : aa_nat_int(power_power_int(A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_3266_power__Suc0__right,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_3267_power__Suc0__right,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_3268_power__Suc0__right,axiom,
    ! [A2: complex] : aa_nat_complex(power_power_complex(A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_3269_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_add
tff(fact_3270_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_add
tff(fact_3271_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_add
tff(fact_3272_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_add
tff(fact_3273_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% add_gr_0
tff(fact_3274_power__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_3275_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( X = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_3276_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_3277_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_3278_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_3279_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_3280_add__neg__numeral__special_I8_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = zero_zero_int ).

% add_neg_numeral_special(8)
tff(fact_3281_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_3282_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_3283_add__neg__numeral__special_I8_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_3284_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_3285_add__neg__numeral__special_I7_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% add_neg_numeral_special(7)
tff(fact_3286_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_3287_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_3288_add__neg__numeral__special_I7_J,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_3289_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_3290_power__strict__increasing__iff,axiom,
    ! [B3: real,X: nat,Y: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_3291_power__strict__increasing__iff,axiom,
    ! [B3: rat,X: nat,Y: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B3),X)),aa_nat_rat(power_power_rat(B3),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_3292_power__strict__increasing__iff,axiom,
    ! [B3: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_3293_power__strict__increasing__iff,axiom,
    ! [B3: int,X: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B3),X)),aa_nat_int(power_power_int(B3),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_3294_power__eq__0__iff,axiom,
    ! [A2: rat,N: nat] :
      ( ( aa_nat_rat(power_power_rat(A2),N) = zero_zero_rat )
    <=> ( ( A2 = zero_zero_rat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_3295_power__eq__0__iff,axiom,
    ! [A2: int,N: nat] :
      ( ( aa_nat_int(power_power_int(A2),N) = zero_zero_int )
    <=> ( ( A2 = zero_zero_int )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_3296_power__eq__0__iff,axiom,
    ! [A2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) = zero_zero_nat )
    <=> ( ( A2 = zero_zero_nat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_3297_power__eq__0__iff,axiom,
    ! [A2: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) = zero_zero_real )
    <=> ( ( A2 = zero_zero_real )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_3298_power__eq__0__iff,axiom,
    ! [A2: complex,N: nat] :
      ( ( aa_nat_complex(power_power_complex(A2),N) = zero_zero_complex )
    <=> ( ( A2 = zero_zero_complex )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_3299_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_3300_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M)) ).

% of_nat_Suc
tff(fact_3301_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_3302_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% of_nat_Suc
tff(fact_3303_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_3304_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_3305_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_3306_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W2: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B3)),W2)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3307_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B3)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3308_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W2: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B3)),W2)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3309_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W2: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B3)),W2)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3310_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A2: real,B3: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_3311_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A2: rat,B3: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_3312_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A2: nat,B3: 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,A2),B3)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_3313_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A2: int,B3: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_3314_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_3315_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_3316_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_3317_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_3318_group__cancel_Oadd1,axiom,
    ! [A3: real,K: real,A2: real,B3: real] :
      ( ( A3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A2) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A3),B3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)) ) ) ).

% group_cancel.add1
tff(fact_3319_group__cancel_Oadd1,axiom,
    ! [A3: rat,K: rat,A2: rat,B3: rat] :
      ( ( A3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A2) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A3),B3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)) ) ) ).

% group_cancel.add1
tff(fact_3320_group__cancel_Oadd1,axiom,
    ! [A3: nat,K: nat,A2: nat,B3: nat] :
      ( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A2) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3)) ) ) ).

% group_cancel.add1
tff(fact_3321_group__cancel_Oadd1,axiom,
    ! [A3: int,K: int,A2: int,B3: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A2) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A3),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)) ) ) ).

% group_cancel.add1
tff(fact_3322_group__cancel_Oadd2,axiom,
    ! [B2: real,K: real,B3: real,A2: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B3) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)) ) ) ).

% group_cancel.add2
tff(fact_3323_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B3: rat,A2: rat] :
      ( ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B3) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)) ) ) ).

% group_cancel.add2
tff(fact_3324_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B3: nat,A2: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B3) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3)) ) ) ).

% group_cancel.add2
tff(fact_3325_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B3: int,A2: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B3) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)) ) ) ).

% group_cancel.add2
tff(fact_3326_add_Oassoc,axiom,
    ! [A2: real,B3: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),C)) ).

% add.assoc
tff(fact_3327_add_Oassoc,axiom,
    ! [A2: rat,B3: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),C)) ).

% add.assoc
tff(fact_3328_add_Oassoc,axiom,
    ! [A2: nat,B3: 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,A2),B3)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),C)) ).

% add.assoc
tff(fact_3329_add_Oassoc,axiom,
    ! [A2: int,B3: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),C)) ).

% add.assoc
tff(fact_3330_add_Oleft__cancel,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C) )
    <=> ( B3 = C ) ) ).

% add.left_cancel
tff(fact_3331_add_Oleft__cancel,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C) )
    <=> ( B3 = C ) ) ).

% add.left_cancel
tff(fact_3332_add_Oleft__cancel,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C) )
    <=> ( B3 = C ) ) ).

% add.left_cancel
tff(fact_3333_add_Oright__cancel,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A2) )
    <=> ( B3 = C ) ) ).

% add.right_cancel
tff(fact_3334_add_Oright__cancel,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2) )
    <=> ( B3 = C ) ) ).

% add.right_cancel
tff(fact_3335_add_Oright__cancel,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2) )
    <=> ( B3 = C ) ) ).

% add.right_cancel
tff(fact_3336_add_Ocommute,axiom,
    ! [A2: real,B3: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),A2) ).

% add.commute
tff(fact_3337_add_Ocommute,axiom,
    ! [A2: rat,B3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),A2) ).

% add.commute
tff(fact_3338_add_Ocommute,axiom,
    ! [A2: nat,B3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),A2) ).

% add.commute
tff(fact_3339_add_Ocommute,axiom,
    ! [A2: int,B3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),A2) ).

% add.commute
tff(fact_3340_add_Oleft__commute,axiom,
    ! [B3: real,A2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),C)) ).

% add.left_commute
tff(fact_3341_add_Oleft__commute,axiom,
    ! [B3: rat,A2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),C)) ).

% add.left_commute
tff(fact_3342_add_Oleft__commute,axiom,
    ! [B3: nat,A2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),C)) ).

% add.left_commute
tff(fact_3343_add_Oleft__commute,axiom,
    ! [B3: int,A2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),C)) ).

% add.left_commute
tff(fact_3344_add__left__imp__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C) )
     => ( B3 = C ) ) ).

% add_left_imp_eq
tff(fact_3345_add__left__imp__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C) )
     => ( B3 = C ) ) ).

% add_left_imp_eq
tff(fact_3346_add__left__imp__eq,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C) )
     => ( B3 = C ) ) ).

% add_left_imp_eq
tff(fact_3347_add__left__imp__eq,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C) )
     => ( B3 = C ) ) ).

% add_left_imp_eq
tff(fact_3348_add__right__imp__eq,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A2) )
     => ( B3 = C ) ) ).

% add_right_imp_eq
tff(fact_3349_add__right__imp__eq,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2) )
     => ( B3 = C ) ) ).

% add_right_imp_eq
tff(fact_3350_add__right__imp__eq,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2) )
     => ( B3 = C ) ) ).

% add_right_imp_eq
tff(fact_3351_add__right__imp__eq,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2) )
     => ( B3 = C ) ) ).

% add_right_imp_eq
tff(fact_3352_add__le__imp__le__right,axiom,
    ! [A2: real,C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),C)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ).

% add_le_imp_le_right
tff(fact_3353_add__le__imp__le__right,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),C)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ).

% add_le_imp_le_right
tff(fact_3354_add__le__imp__le__right,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),C)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3)) ) ).

% add_le_imp_le_right
tff(fact_3355_add__le__imp__le__right,axiom,
    ! [A2: int,C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),C)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ).

% add_le_imp_le_right
tff(fact_3356_add__le__imp__le__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B3)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ).

% add_le_imp_le_left
tff(fact_3357_add__le__imp__le__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B3)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ).

% add_le_imp_le_left
tff(fact_3358_add__le__imp__le__left,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B3)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3)) ) ).

% add_le_imp_le_left
tff(fact_3359_add__le__imp__le__left,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B3)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ).

% add_le_imp_le_left
tff(fact_3360_le__iff__add,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
    <=> ? [C5: nat] : B3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C5) ) ).

% le_iff_add
tff(fact_3361_add__right__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),C))) ) ).

% add_right_mono
tff(fact_3362_add__right__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),C))) ) ).

% add_right_mono
tff(fact_3363_add__right__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),C))) ) ).

% add_right_mono
tff(fact_3364_add__right__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),C))) ) ).

% add_right_mono
tff(fact_3365_less__eqE,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ~ ! [C3: nat] : B3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C3) ) ).

% less_eqE
tff(fact_3366_add__left__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B3))) ) ).

% add_left_mono
tff(fact_3367_add__left__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B3))) ) ).

% add_left_mono
tff(fact_3368_add__left__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B3))) ) ).

% add_left_mono
tff(fact_3369_add__left__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B3))) ) ).

% add_left_mono
tff(fact_3370_add__mono,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),D))) ) ) ).

% add_mono
tff(fact_3371_add__mono,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),D))) ) ) ).

% add_mono
tff(fact_3372_add__mono,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),D))) ) ) ).

% add_mono
tff(fact_3373_add__mono,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),D))) ) ) ).

% add_mono
tff(fact_3374_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3375_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3376_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3377_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3378_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3379_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3380_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3381_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3382_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & ( K = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3383_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & ( K = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3384_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & ( K = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3385_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & ( K = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3386_verit__sum__simplify,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),zero_zero_real) = A2 ).

% verit_sum_simplify
tff(fact_3387_verit__sum__simplify,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),zero_zero_rat) = A2 ).

% verit_sum_simplify
tff(fact_3388_verit__sum__simplify,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),zero_zero_nat) = A2 ).

% verit_sum_simplify
tff(fact_3389_verit__sum__simplify,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),zero_zero_int) = A2 ).

% verit_sum_simplify
tff(fact_3390_add_Ogroup__left__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A2) = A2 ).

% add.group_left_neutral
tff(fact_3391_add_Ogroup__left__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A2) = A2 ).

% add.group_left_neutral
tff(fact_3392_add_Ogroup__left__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A2) = A2 ).

% add.group_left_neutral
tff(fact_3393_add_Ocomm__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),zero_zero_real) = A2 ).

% add.comm_neutral
tff(fact_3394_add_Ocomm__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),zero_zero_rat) = A2 ).

% add.comm_neutral
tff(fact_3395_add_Ocomm__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),zero_zero_nat) = A2 ).

% add.comm_neutral
tff(fact_3396_add_Ocomm__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),zero_zero_int) = A2 ).

% add.comm_neutral
tff(fact_3397_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_3398_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_3399_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_3400_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_3401_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3402_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3403_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3404_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3405_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3406_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3407_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3408_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3409_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & ( K = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3410_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & ( K = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3411_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( K = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3412_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & ( K = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3413_add__strict__mono,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),D))) ) ) ).

% add_strict_mono
tff(fact_3414_add__strict__mono,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),D))) ) ) ).

% add_strict_mono
tff(fact_3415_add__strict__mono,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),D))) ) ) ).

% add_strict_mono
tff(fact_3416_add__strict__mono,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),D))) ) ) ).

% add_strict_mono
tff(fact_3417_add__strict__left__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B3))) ) ).

% add_strict_left_mono
tff(fact_3418_add__strict__left__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B3))) ) ).

% add_strict_left_mono
tff(fact_3419_add__strict__left__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B3))) ) ).

% add_strict_left_mono
tff(fact_3420_add__strict__left__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B3))) ) ).

% add_strict_left_mono
tff(fact_3421_add__strict__right__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),C))) ) ).

% add_strict_right_mono
tff(fact_3422_add__strict__right__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),C))) ) ).

% add_strict_right_mono
tff(fact_3423_add__strict__right__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),C))) ) ).

% add_strict_right_mono
tff(fact_3424_add__strict__right__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),C))) ) ).

% add_strict_right_mono
tff(fact_3425_add__less__imp__less__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B3)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) ) ).

% add_less_imp_less_left
tff(fact_3426_add__less__imp__less__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B3)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) ) ).

% add_less_imp_less_left
tff(fact_3427_add__less__imp__less__left,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B3)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3)) ) ).

% add_less_imp_less_left
tff(fact_3428_add__less__imp__less__left,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B3)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3)) ) ).

% add_less_imp_less_left
tff(fact_3429_add__less__imp__less__right,axiom,
    ! [A2: real,C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),C)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) ) ).

% add_less_imp_less_right
tff(fact_3430_add__less__imp__less__right,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),C)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) ) ).

% add_less_imp_less_right
tff(fact_3431_add__less__imp__less__right,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),C)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3)) ) ).

% add_less_imp_less_right
tff(fact_3432_add__less__imp__less__right,axiom,
    ! [A2: int,C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),C)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3)) ) ).

% add_less_imp_less_right
tff(fact_3433_group__cancel_Osub1,axiom,
    ! [A3: real,K: real,A2: real,B3: real] :
      ( ( A3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A2) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A3),B3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)) ) ) ).

% group_cancel.sub1
tff(fact_3434_group__cancel_Osub1,axiom,
    ! [A3: rat,K: rat,A2: rat,B3: rat] :
      ( ( A3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A2) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A3),B3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)) ) ) ).

% group_cancel.sub1
tff(fact_3435_group__cancel_Osub1,axiom,
    ! [A3: int,K: int,A2: int,B3: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A2) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A3),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)) ) ) ).

% group_cancel.sub1
tff(fact_3436_diff__eq__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3) = C )
    <=> ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B3) ) ) ).

% diff_eq_eq
tff(fact_3437_diff__eq__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3) = C )
    <=> ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B3) ) ) ).

% diff_eq_eq
tff(fact_3438_diff__eq__eq,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3) = C )
    <=> ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B3) ) ) ).

% diff_eq_eq
tff(fact_3439_eq__diff__eq,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B3) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3) = C ) ) ).

% eq_diff_eq
tff(fact_3440_eq__diff__eq,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B3) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3) = C ) ) ).

% eq_diff_eq
tff(fact_3441_eq__diff__eq,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B3) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3) = C ) ) ).

% eq_diff_eq
tff(fact_3442_add__diff__eq,axiom,
    ! [A2: real,B3: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),C) ).

% add_diff_eq
tff(fact_3443_add__diff__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B3),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)),C) ).

% add_diff_eq
tff(fact_3444_add__diff__eq,axiom,
    ! [A2: int,B3: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,B3),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)),C) ).

% add_diff_eq
tff(fact_3445_diff__diff__eq2,axiom,
    ! [A2: real,B3: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),B3) ).

% diff_diff_eq2
tff(fact_3446_diff__diff__eq2,axiom,
    ! [A2: rat,B3: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B3),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),B3) ).

% diff_diff_eq2
tff(fact_3447_diff__diff__eq2,axiom,
    ! [A2: int,B3: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,B3),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),B3) ).

% diff_diff_eq2
tff(fact_3448_diff__add__eq,axiom,
    ! [A2: real,B3: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),B3) ).

% diff_add_eq
tff(fact_3449_diff__add__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),B3) ).

% diff_add_eq
tff(fact_3450_diff__add__eq,axiom,
    ! [A2: int,B3: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),B3) ).

% diff_add_eq
tff(fact_3451_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: real,B3: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C)),B3) ).

% diff_add_eq_diff_diff_swap
tff(fact_3452_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: rat,B3: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),C)),B3) ).

% diff_add_eq_diff_diff_swap
tff(fact_3453_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: int,B3: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C)),B3) ).

% diff_add_eq_diff_diff_swap
tff(fact_3454_add__implies__diff,axiom,
    ! [C: real,B3: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B3) = A2 )
     => ( C = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3) ) ) ).

% add_implies_diff
tff(fact_3455_add__implies__diff,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B3) = A2 )
     => ( C = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3) ) ) ).

% add_implies_diff
tff(fact_3456_add__implies__diff,axiom,
    ! [C: nat,B3: nat,A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B3) = A2 )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B3) ) ) ).

% add_implies_diff
tff(fact_3457_add__implies__diff,axiom,
    ! [C: int,B3: int,A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B3) = A2 )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3) ) ) ).

% add_implies_diff
tff(fact_3458_diff__diff__eq,axiom,
    ! [A2: real,B3: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),C)) ).

% diff_diff_eq
tff(fact_3459_diff__diff__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),C)) ).

% diff_diff_eq
tff(fact_3460_diff__diff__eq,axiom,
    ! [A2: nat,B3: 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,A2),B3)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),C)) ).

% diff_diff_eq
tff(fact_3461_diff__diff__eq,axiom,
    ! [A2: int,B3: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),C)) ).

% diff_diff_eq
tff(fact_3462_group__cancel_Oneg1,axiom,
    ! [A3: int,K: int,A2: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A2) )
     => ( aa_int_int(uminus_uminus_int,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A2)) ) ) ).

% group_cancel.neg1
tff(fact_3463_group__cancel_Oneg1,axiom,
    ! [A3: real,K: real,A2: real] :
      ( ( A3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A2) )
     => ( aa_real_real(uminus_uminus_real,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A2)) ) ) ).

% group_cancel.neg1
tff(fact_3464_group__cancel_Oneg1,axiom,
    ! [A3: rat,K: rat,A2: rat] :
      ( ( A3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A2) )
     => ( aa_rat_rat(uminus_uminus_rat,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A2)) ) ) ).

% group_cancel.neg1
tff(fact_3465_group__cancel_Oneg1,axiom,
    ! [A3: code_integer,K: code_integer,A2: code_integer] :
      ( ( A3 = plus_p5714425477246183910nteger(K,A2) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A3) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)) ) ) ).

% group_cancel.neg1
tff(fact_3466_group__cancel_Oneg1,axiom,
    ! [A3: complex,K: complex,A2: complex] :
      ( ( A3 = aa_complex_complex(plus_plus_complex(K),A2) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A3) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,A2)) ) ) ).

% group_cancel.neg1
tff(fact_3467_add_Oinverse__distrib__swap,axiom,
    ! [A2: int,B3: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B3)),aa_int_int(uminus_uminus_int,A2)) ).

% add.inverse_distrib_swap
tff(fact_3468_add_Oinverse__distrib__swap,axiom,
    ! [A2: real,B3: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B3)),aa_real_real(uminus_uminus_real,A2)) ).

% add.inverse_distrib_swap
tff(fact_3469_add_Oinverse__distrib__swap,axiom,
    ! [A2: rat,B3: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,B3)),aa_rat_rat(uminus_uminus_rat,A2)) ).

% add.inverse_distrib_swap
tff(fact_3470_add_Oinverse__distrib__swap,axiom,
    ! [A2: code_integer,B3: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A2,B3)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)) ).

% add.inverse_distrib_swap
tff(fact_3471_add_Oinverse__distrib__swap,axiom,
    ! [A2: complex,B3: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A2),B3)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B3)),aa_complex_complex(uminus1482373934393186551omplex,A2)) ).

% add.inverse_distrib_swap
tff(fact_3472_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A2: nat] :
      ( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A2) )
     => ( aa_nat_nat(suc,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A2)) ) ) ).

% nat_arith.suc1
tff(fact_3473_add__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% add_Suc
tff(fact_3474_add__Suc__shift,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) ).

% add_Suc_shift
tff(fact_3475_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_3476_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N) = N ).

% plus_nat.add_0
tff(fact_3477_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).

% add_lessD1
tff(fact_3478_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).

% add_less_mono
tff(fact_3479_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I)) ).

% not_add_less1
tff(fact_3480_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I)) ).

% not_add_less2
tff(fact_3481_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).

% add_less_mono1
tff(fact_3482_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).

% trans_less_add1
tff(fact_3483_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).

% trans_less_add2
tff(fact_3484_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% less_add_eq_less
tff(fact_3485_nat__le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> ? [K3: nat] : N = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3) ) ).

% nat_le_iff_add
tff(fact_3486_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).

% trans_le_add2
tff(fact_3487_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).

% trans_le_add1
tff(fact_3488_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).

% add_le_mono1
tff(fact_3489_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).

% add_le_mono
tff(fact_3490_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
     => ? [N2: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2) ) ).

% le_Suc_ex
tff(fact_3491_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).

% add_leD2
tff(fact_3492_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% add_leD1
tff(fact_3493_le__add2,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% le_add2
tff(fact_3494_le__add1,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))) ).

% le_add1
tff(fact_3495_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ) ).

% add_leE
tff(fact_3496_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% Nat.diff_cancel
tff(fact_3497_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% diff_cancel2
tff(fact_3498_diff__add__inverse,axiom,
    ! [N: 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,N),M)),N) = M ).

% diff_add_inverse
tff(fact_3499_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = M ).

% diff_add_inverse2
tff(fact_3500_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_3501_abs__real__def,axiom,
    ! [A2: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A2) = aa_real_real(uminus_uminus_real,A2) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A2) = A2 ) ) ) ).

% abs_real_def
tff(fact_3502_add__decreasing,axiom,
    ! [A2: real,C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),B3)) ) ) ).

% add_decreasing
tff(fact_3503_add__decreasing,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),B3)) ) ) ).

% add_decreasing
tff(fact_3504_add__decreasing,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),B3)) ) ) ).

% add_decreasing
tff(fact_3505_add__decreasing,axiom,
    ! [A2: int,C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),B3)) ) ) ).

% add_decreasing
tff(fact_3506_add__increasing,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C))) ) ) ).

% add_increasing
tff(fact_3507_add__increasing,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C))) ) ) ).

% add_increasing
tff(fact_3508_add__increasing,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C))) ) ) ).

% add_increasing
tff(fact_3509_add__increasing,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C))) ) ) ).

% add_increasing
tff(fact_3510_add__decreasing2,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),B3)) ) ) ).

% add_decreasing2
tff(fact_3511_add__decreasing2,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),B3)) ) ) ).

% add_decreasing2
tff(fact_3512_add__decreasing2,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),B3)) ) ) ).

% add_decreasing2
tff(fact_3513_add__decreasing2,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),B3)) ) ) ).

% add_decreasing2
tff(fact_3514_add__increasing2,axiom,
    ! [C: real,B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C))) ) ) ).

% add_increasing2
tff(fact_3515_add__increasing2,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C))) ) ) ).

% add_increasing2
tff(fact_3516_add__increasing2,axiom,
    ! [C: nat,B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C))) ) ) ).

% add_increasing2
tff(fact_3517_add__increasing2,axiom,
    ! [C: int,B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C))) ) ) ).

% add_increasing2
tff(fact_3518_add__nonneg__nonneg,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3))) ) ) ).

% add_nonneg_nonneg
tff(fact_3519_add__nonneg__nonneg,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3))) ) ) ).

% add_nonneg_nonneg
tff(fact_3520_add__nonneg__nonneg,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3))) ) ) ).

% add_nonneg_nonneg
tff(fact_3521_add__nonneg__nonneg,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3))) ) ) ).

% add_nonneg_nonneg
tff(fact_3522_add__nonpos__nonpos,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),zero_zero_real)) ) ) ).

% add_nonpos_nonpos
tff(fact_3523_add__nonpos__nonpos,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)),zero_zero_rat)) ) ) ).

% add_nonpos_nonpos
tff(fact_3524_add__nonpos__nonpos,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3)),zero_zero_nat)) ) ) ).

% add_nonpos_nonpos
tff(fact_3525_add__nonpos__nonpos,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)),zero_zero_int)) ) ) ).

% add_nonpos_nonpos
tff(fact_3526_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3527_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3528_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3529_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3530_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3531_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3532_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),zero_zero_nat))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3533_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3534_add__less__le__mono,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),D))) ) ) ).

% add_less_le_mono
tff(fact_3535_add__less__le__mono,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),D))) ) ) ).

% add_less_le_mono
tff(fact_3536_add__less__le__mono,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),D))) ) ) ).

% add_less_le_mono
tff(fact_3537_add__less__le__mono,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),D))) ) ) ).

% add_less_le_mono
tff(fact_3538_add__le__less__mono,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),D))) ) ) ).

% add_le_less_mono
tff(fact_3539_add__le__less__mono,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),D))) ) ) ).

% add_le_less_mono
tff(fact_3540_add__le__less__mono,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),D))) ) ) ).

% add_le_less_mono
tff(fact_3541_add__le__less__mono,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),D))) ) ) ).

% add_le_less_mono
tff(fact_3542_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3543_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3544_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3545_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3546_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3547_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3548_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3549_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3550_pos__add__strict,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C))) ) ) ).

% pos_add_strict
tff(fact_3551_pos__add__strict,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C))) ) ) ).

% pos_add_strict
tff(fact_3552_pos__add__strict,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C))) ) ) ).

% pos_add_strict
tff(fact_3553_pos__add__strict,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C))) ) ) ).

% pos_add_strict
tff(fact_3554_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ~ ! [C3: nat] :
            ( ( B3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C3) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_3555_add__pos__pos,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3))) ) ) ).

% add_pos_pos
tff(fact_3556_add__pos__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3))) ) ) ).

% add_pos_pos
tff(fact_3557_add__pos__pos,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3))) ) ) ).

% add_pos_pos
tff(fact_3558_add__pos__pos,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3))) ) ) ).

% add_pos_pos
tff(fact_3559_add__neg__neg,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),zero_zero_real)) ) ) ).

% add_neg_neg
tff(fact_3560_add__neg__neg,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)),zero_zero_rat)) ) ) ).

% add_neg_neg
tff(fact_3561_add__neg__neg,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3)),zero_zero_nat)) ) ) ).

% add_neg_neg
tff(fact_3562_add__neg__neg,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)),zero_zero_int)) ) ) ).

% add_neg_neg
tff(fact_3563_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real)) ) ) ).

% add_less_zeroD
tff(fact_3564_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat)) ) ) ).

% add_less_zeroD
tff(fact_3565_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),zero_zero_int)) ) ) ).

% add_less_zeroD
tff(fact_3566_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),K))) ) ).

% power_gt_expt
tff(fact_3567_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B3),A2) = C )
        <=> ( B3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_3568_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B3),A2)) = B3 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_3569_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B3),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),B3) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_3570_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),C)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B3),A2)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_3571_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B3),A2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),C)),A2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_3572_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B3)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B3),A2)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_3573_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B3),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B3)),A2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_3574_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B3),A2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),B3)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_3575_le__add__diff,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),C)),A2))) ) ).

% le_add_diff
tff(fact_3576_diff__add,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B3),A2)),A2) = B3 ) ) ).

% diff_add
tff(fact_3577_le__diff__eq,axiom,
    ! [A2: real,C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),C)) ) ).

% le_diff_eq
tff(fact_3578_le__diff__eq,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B3)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)),C)) ) ).

% le_diff_eq
tff(fact_3579_le__diff__eq,axiom,
    ! [A2: int,C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)),C)) ) ).

% le_diff_eq
tff(fact_3580_diff__le__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)),C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B3))) ) ).

% diff_le_eq
tff(fact_3581_diff__le__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B3))) ) ).

% diff_le_eq
tff(fact_3582_diff__le__eq,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B3))) ) ).

% diff_le_eq
tff(fact_3583_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N: real,J: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3584_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N: rat,J: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K)))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3585_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3586_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3587_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_3588_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_3589_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_3590_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_3591_less__add__one,axiom,
    ! [A2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),one_one_real))) ).

% less_add_one
tff(fact_3592_less__add__one,axiom,
    ! [A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),one_one_rat))) ).

% less_add_one
tff(fact_3593_less__add__one,axiom,
    ! [A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat))) ).

% less_add_one
tff(fact_3594_less__add__one,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int))) ).

% less_add_one
tff(fact_3595_add__mono1,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),one_one_real))) ) ).

% add_mono1
tff(fact_3596_add__mono1,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),one_one_rat))) ) ).

% add_mono1
tff(fact_3597_add__mono1,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),one_one_nat))) ) ).

% add_mono1
tff(fact_3598_add__mono1,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),one_one_int))) ) ).

% add_mono1
tff(fact_3599_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N))) ) ).

% nat_one_le_power
tff(fact_3600_less__diff__eq,axiom,
    ! [A2: real,C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),C)) ) ).

% less_diff_eq
tff(fact_3601_less__diff__eq,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B3)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)),C)) ) ).

% less_diff_eq
tff(fact_3602_less__diff__eq,axiom,
    ! [A2: int,C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)),C)) ) ).

% less_diff_eq
tff(fact_3603_diff__less__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)),C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B3))) ) ).

% diff_less_eq
tff(fact_3604_diff__less__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B3))) ) ).

% diff_less_eq
tff(fact_3605_diff__less__eq,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B3))) ) ).

% diff_less_eq
tff(fact_3606_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: real,B3: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3607_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: rat,B3: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3608_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B3: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B3)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3609_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: int,B3: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3610_add__eq__0__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3) = zero_zero_int )
    <=> ( B3 = aa_int_int(uminus_uminus_int,A2) ) ) ).

% add_eq_0_iff
tff(fact_3611_add__eq__0__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3) = zero_zero_real )
    <=> ( B3 = aa_real_real(uminus_uminus_real,A2) ) ) ).

% add_eq_0_iff
tff(fact_3612_add__eq__0__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3) = zero_zero_rat )
    <=> ( B3 = aa_rat_rat(uminus_uminus_rat,A2) ) ) ).

% add_eq_0_iff
tff(fact_3613_add__eq__0__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( plus_p5714425477246183910nteger(A2,B3) = zero_z3403309356797280102nteger )
    <=> ( B3 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) ) ) ).

% add_eq_0_iff
tff(fact_3614_add__eq__0__iff,axiom,
    ! [A2: complex,B3: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A2),B3) = zero_zero_complex )
    <=> ( B3 = aa_complex_complex(uminus1482373934393186551omplex,A2) ) ) ).

% add_eq_0_iff
tff(fact_3615_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A2)),A2) = zero_zero_int ).

% ab_group_add_class.ab_left_minus
tff(fact_3616_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),A2) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_3617_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A2)),A2) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_3618_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2),A2) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_3619_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),A2) = zero_zero_complex ).

% ab_group_add_class.ab_left_minus
tff(fact_3620_add_Oinverse__unique,axiom,
    ! [A2: int,B3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A2) = B3 ) ) ).

% add.inverse_unique
tff(fact_3621_add_Oinverse__unique,axiom,
    ! [A2: real,B3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A2) = B3 ) ) ).

% add.inverse_unique
tff(fact_3622_add_Oinverse__unique,axiom,
    ! [A2: rat,B3: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A2) = B3 ) ) ).

% add.inverse_unique
tff(fact_3623_add_Oinverse__unique,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( plus_p5714425477246183910nteger(A2,B3) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = B3 ) ) ).

% add.inverse_unique
tff(fact_3624_add_Oinverse__unique,axiom,
    ! [A2: complex,B3: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A2),B3) = zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A2) = B3 ) ) ).

% add.inverse_unique
tff(fact_3625_eq__neg__iff__add__eq__0,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 = aa_int_int(uminus_uminus_int,B3) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_3626_eq__neg__iff__add__eq__0,axiom,
    ! [A2: real,B3: real] :
      ( ( A2 = aa_real_real(uminus_uminus_real,B3) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_3627_eq__neg__iff__add__eq__0,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 = aa_rat_rat(uminus_uminus_rat,B3) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_3628_eq__neg__iff__add__eq__0,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3) )
    <=> ( plus_p5714425477246183910nteger(A2,B3) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_3629_eq__neg__iff__add__eq__0,axiom,
    ! [A2: complex,B3: complex] :
      ( ( A2 = aa_complex_complex(uminus1482373934393186551omplex,B3) )
    <=> ( aa_complex_complex(plus_plus_complex(A2),B3) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_3630_neg__eq__iff__add__eq__0,axiom,
    ! [A2: int,B3: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = B3 )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_3631_neg__eq__iff__add__eq__0,axiom,
    ! [A2: real,B3: real] :
      ( ( aa_real_real(uminus_uminus_real,A2) = B3 )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_3632_neg__eq__iff__add__eq__0,axiom,
    ! [A2: rat,B3: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A2) = B3 )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_3633_neg__eq__iff__add__eq__0,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = B3 )
    <=> ( plus_p5714425477246183910nteger(A2,B3) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_3634_neg__eq__iff__add__eq__0,axiom,
    ! [A2: complex,B3: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A2) = B3 )
    <=> ( aa_complex_complex(plus_plus_complex(A2),B3) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_3635_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B3: int,A2: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B3) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)) ) ) ).

% group_cancel.sub2
tff(fact_3636_group__cancel_Osub2,axiom,
    ! [B2: real,K: real,B3: real,A2: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B3) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)) ) ) ).

% group_cancel.sub2
tff(fact_3637_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B3: rat,A2: rat] :
      ( ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B3) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)) ) ) ).

% group_cancel.sub2
tff(fact_3638_group__cancel_Osub2,axiom,
    ! [B2: code_integer,K: code_integer,B3: code_integer,A2: code_integer] :
      ( ( B2 = plus_p5714425477246183910nteger(K,B3) )
     => ( minus_8373710615458151222nteger(A2,B2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),minus_8373710615458151222nteger(A2,B3)) ) ) ).

% group_cancel.sub2
tff(fact_3639_group__cancel_Osub2,axiom,
    ! [B2: complex,K: complex,B3: complex,A2: complex] :
      ( ( B2 = aa_complex_complex(plus_plus_complex(K),B3) )
     => ( aa_complex_complex(minus_minus_complex(A2),B2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(minus_minus_complex(A2),B3)) ) ) ).

% group_cancel.sub2
tff(fact_3640_diff__conv__add__uminus,axiom,
    ! [A2: int,B3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(uminus_uminus_int,B3)) ).

% diff_conv_add_uminus
tff(fact_3641_diff__conv__add__uminus,axiom,
    ! [A2: real,B3: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(uminus_uminus_real,B3)) ).

% diff_conv_add_uminus
tff(fact_3642_diff__conv__add__uminus,axiom,
    ! [A2: rat,B3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(uminus_uminus_rat,B3)) ).

% diff_conv_add_uminus
tff(fact_3643_diff__conv__add__uminus,axiom,
    ! [A2: code_integer,B3: code_integer] : minus_8373710615458151222nteger(A2,B3) = plus_p5714425477246183910nteger(A2,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3)) ).

% diff_conv_add_uminus
tff(fact_3644_diff__conv__add__uminus,axiom,
    ! [A2: complex,B3: complex] : aa_complex_complex(minus_minus_complex(A2),B3) = aa_complex_complex(plus_plus_complex(A2),aa_complex_complex(uminus1482373934393186551omplex,B3)) ).

% diff_conv_add_uminus
tff(fact_3645_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A2: int,B3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(uminus_uminus_int,B3)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3646_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A2: real,B3: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(uminus_uminus_real,B3)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3647_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A2: rat,B3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(uminus_uminus_rat,B3)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3648_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A2: code_integer,B3: code_integer] : minus_8373710615458151222nteger(A2,B3) = plus_p5714425477246183910nteger(A2,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B3)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3649_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A2: complex,B3: complex] : aa_complex_complex(minus_minus_complex(A2),B3) = aa_complex_complex(plus_plus_complex(A2),aa_complex_complex(uminus1482373934393186551omplex,B3)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3650_abs__triangle__ineq,axiom,
    ! [A2: code_integer,B3: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(A2,B3))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B3)))) ).

% abs_triangle_ineq
tff(fact_3651_abs__triangle__ineq,axiom,
    ! [A2: real,B3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B3)))) ).

% abs_triangle_ineq
tff(fact_3652_abs__triangle__ineq,axiom,
    ! [A2: rat,B3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B3)))) ).

% abs_triangle_ineq
tff(fact_3653_abs__triangle__ineq,axiom,
    ! [A2: int,B3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B3)))) ).

% abs_triangle_ineq
tff(fact_3654_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_3655_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_3656_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ? [K2: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).

% less_imp_Suc_add
tff(fact_3657_less__iff__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> ? [K3: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).

% less_iff_Suc_add
tff(fact_3658_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),I)))) ).

% less_add_Suc2
tff(fact_3659_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)))) ).

% less_add_Suc1
tff(fact_3660_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ~ ! [Q4: nat] : N != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)) ) ).

% less_natE
tff(fact_3661_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N2))) ) ).

% real_arch_pow
tff(fact_3662_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ? [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_3663_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M4: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,M4)),aa_nat_nat(F,N2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(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_3664_diff__add__0,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_3665_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_3666_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ).

% less_diff_conv
tff(fact_3667_Suc__eq__plus1,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ).

% Suc_eq_plus1
tff(fact_3668_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_3669_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ).

% Suc_eq_plus1_left
tff(fact_3670_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_3671_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_3672_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_3673_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ) ).

% Nat.le_diff_conv2
tff(fact_3674_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ).

% le_diff_conv
tff(fact_3675_add__strict__increasing2,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C))) ) ) ).

% add_strict_increasing2
tff(fact_3676_add__strict__increasing2,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C))) ) ) ).

% add_strict_increasing2
tff(fact_3677_add__strict__increasing2,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C))) ) ) ).

% add_strict_increasing2
tff(fact_3678_add__strict__increasing2,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C))) ) ) ).

% add_strict_increasing2
tff(fact_3679_add__strict__increasing,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C))) ) ) ).

% add_strict_increasing
tff(fact_3680_add__strict__increasing,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C))) ) ) ).

% add_strict_increasing
tff(fact_3681_add__strict__increasing,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C))) ) ) ).

% add_strict_increasing
tff(fact_3682_add__strict__increasing,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C))) ) ) ).

% add_strict_increasing
tff(fact_3683_add__pos__nonneg,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3))) ) ) ).

% add_pos_nonneg
tff(fact_3684_add__pos__nonneg,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3))) ) ) ).

% add_pos_nonneg
tff(fact_3685_add__pos__nonneg,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3))) ) ) ).

% add_pos_nonneg
tff(fact_3686_add__pos__nonneg,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3))) ) ) ).

% add_pos_nonneg
tff(fact_3687_add__nonpos__neg,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),zero_zero_real)) ) ) ).

% add_nonpos_neg
tff(fact_3688_add__nonpos__neg,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)),zero_zero_rat)) ) ) ).

% add_nonpos_neg
tff(fact_3689_add__nonpos__neg,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3)),zero_zero_nat)) ) ) ).

% add_nonpos_neg
tff(fact_3690_add__nonpos__neg,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)),zero_zero_int)) ) ) ).

% add_nonpos_neg
tff(fact_3691_add__nonneg__pos,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3))) ) ) ).

% add_nonneg_pos
tff(fact_3692_add__nonneg__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3))) ) ) ).

% add_nonneg_pos
tff(fact_3693_add__nonneg__pos,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3))) ) ) ).

% add_nonneg_pos
tff(fact_3694_add__nonneg__pos,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3))) ) ) ).

% add_nonneg_pos
tff(fact_3695_add__neg__nonpos,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),zero_zero_real)) ) ) ).

% add_neg_nonpos
tff(fact_3696_add__neg__nonpos,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)),zero_zero_rat)) ) ) ).

% add_neg_nonpos
tff(fact_3697_add__neg__nonpos,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3)),zero_zero_nat)) ) ) ).

% add_neg_nonpos
tff(fact_3698_add__neg__nonpos,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)),zero_zero_int)) ) ) ).

% add_neg_nonpos
tff(fact_3699_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_epsilon
tff(fact_3700_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),E))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% field_le_epsilon
tff(fact_3701_discrete,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),B3)) ) ).

% discrete
tff(fact_3702_discrete,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int)),B3)) ) ).

% discrete
tff(fact_3703_zero__less__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ).

% zero_less_two
tff(fact_3704_zero__less__two,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ).

% zero_less_two
tff(fact_3705_zero__less__two,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% zero_less_two
tff(fact_3706_zero__less__two,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int))) ).

% zero_less_two
tff(fact_3707_div__add__self2,axiom,
    ! [B3: int,A2: int] :
      ( ( B3 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)),one_one_int) ) ) ).

% div_add_self2
tff(fact_3708_div__add__self2,axiom,
    ! [B3: nat,A2: nat] :
      ( ( B3 != 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,A2),B3)),B3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B3)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_3709_div__add__self1,axiom,
    ! [B3: int,A2: int] :
      ( ( B3 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),A2)),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)),one_one_int) ) ) ).

% div_add_self1
tff(fact_3710_div__add__self1,axiom,
    ! [B3: nat,A2: nat] :
      ( ( B3 != 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,B3),A2)),B3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B3)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_3711_gt__half__sum,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))),B3)) ) ).

% gt_half_sum
tff(fact_3712_gt__half__sum,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B3)) ) ).

% gt_half_sum
tff(fact_3713_less__half__sum,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)))) ) ).

% less_half_sum
tff(fact_3714_less__half__sum,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real)))) ) ).

% less_half_sum
tff(fact_3715_abs__diff__le__iff,axiom,
    ! [X: code_integer,A2: code_integer,R2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A2))),R2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A2,R2)),X))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),plus_p5714425477246183910nteger(A2,R2))) ) ) ).

% abs_diff_le_iff
tff(fact_3716_abs__diff__le__iff,axiom,
    ! [X: real,A2: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A2))),R2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),R2)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_3717_abs__diff__le__iff,axiom,
    ! [X: rat,A2: rat,R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),A2))),R2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),R2)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_3718_abs__diff__le__iff,axiom,
    ! [X: int,A2: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A2))),R2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),R2)),X))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_3719_abs__diff__triangle__ineq,axiom,
    ! [A2: code_integer,B3: code_integer,C: code_integer,D: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A2,B3),plus_p5714425477246183910nteger(C,D)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A2,C)),abs_abs_Code_integer(minus_8373710615458151222nteger(B3,D))))) ).

% abs_diff_triangle_ineq
tff(fact_3720_abs__diff__triangle__ineq,axiom,
    ! [A2: real,B3: real,C: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),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,A2),C))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),D))))) ).

% abs_diff_triangle_ineq
tff(fact_3721_abs__diff__triangle__ineq,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),D)))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B3),D))))) ).

% abs_diff_triangle_ineq
tff(fact_3722_abs__diff__triangle__ineq,axiom,
    ! [A2: int,B3: int,C: int,D: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),D)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B3),D))))) ).

% abs_diff_triangle_ineq
tff(fact_3723_abs__triangle__ineq4,axiom,
    ! [A2: code_integer,B3: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A2,B3))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B3)))) ).

% abs_triangle_ineq4
tff(fact_3724_abs__triangle__ineq4,axiom,
    ! [A2: real,B3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B3)))) ).

% abs_triangle_ineq4
tff(fact_3725_abs__triangle__ineq4,axiom,
    ! [A2: rat,B3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B3)))) ).

% abs_triangle_ineq4
tff(fact_3726_abs__triangle__ineq4,axiom,
    ! [A2: int,B3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B3)))) ).

% abs_triangle_ineq4
tff(fact_3727_abs__diff__less__iff,axiom,
    ! [X: code_integer,A2: code_integer,R2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A2))),R2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A2,R2)),X))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),plus_p5714425477246183910nteger(A2,R2))) ) ) ).

% abs_diff_less_iff
tff(fact_3728_abs__diff__less__iff,axiom,
    ! [X: real,A2: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A2))),R2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),R2)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_3729_abs__diff__less__iff,axiom,
    ! [X: rat,A2: rat,R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),A2))),R2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),R2)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_3730_abs__diff__less__iff,axiom,
    ! [X: int,A2: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A2))),R2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),R2)),X))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_3731_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N2)),Y)) ) ) ).

% real_arch_pow_inv
tff(fact_3732_nat__diff__split__asm,axiom,
    ! [P: fun_nat_bool,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B3)))
    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
            & ~ pp(aa_nat_bool(P,zero_zero_nat)) )
          | ? [D5: nat] :
              ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),D5) )
              & ~ pp(aa_nat_bool(P,D5)) ) ) ) ).

% nat_diff_split_asm
tff(fact_3733_nat__diff__split,axiom,
    ! [P: fun_nat_bool,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B3)))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ! [D5: nat] :
            ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),D5) )
           => pp(aa_nat_bool(P,D5)) ) ) ) ).

% nat_diff_split
tff(fact_3734_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ) ).

% less_diff_conv2
tff(fact_3735_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(X)))) ).

% abs_add_one_gt_zero
tff(fact_3736_abs__add__one__gt__zero,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(abs_abs_real,X)))) ).

% abs_add_one_gt_zero
tff(fact_3737_abs__add__one__gt__zero,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(abs_abs_rat,X)))) ).

% abs_add_one_gt_zero
tff(fact_3738_abs__add__one__gt__zero,axiom,
    ! [X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(abs_abs_int,X)))) ).

% abs_add_one_gt_zero
tff(fact_3739_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).

% add_eq_if
tff(fact_3740_frac__add,axiom,
    ! [X: real,Y: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real) ) ) ) ).

% frac_add
tff(fact_3741_frac__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat) ) ) ) ).

% frac_add
tff(fact_3742_option_Osize__gen_I2_J,axiom,
    ! [X: fun_nat_nat,X2: nat] : aa_option_nat_nat(size_option_nat(X),some_nat(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(X,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_3743_option_Osize__gen_I2_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat,X2: product_prod_nat_nat] : size_o8335143837870341156at_nat(X,some_P7363390416028606310at_nat(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(X,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_3744_option_Osize__gen_I2_J,axiom,
    ! [X: fun_num_nat,X2: num] : size_option_num(X,some_num(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(X,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_3745_power__not__zero,axiom,
    ! [A2: rat,N: nat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A2),N) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_3746_power__not__zero,axiom,
    ! [A2: int,N: nat] :
      ( ( A2 != zero_zero_int )
     => ( aa_nat_int(power_power_int(A2),N) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_3747_power__not__zero,axiom,
    ! [A2: nat,N: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_3748_power__not__zero,axiom,
    ! [A2: real,N: nat] :
      ( ( A2 != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_3749_power__not__zero,axiom,
    ! [A2: complex,N: nat] :
      ( ( A2 != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A2),N) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_3750_nat0__intermed__int__val,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat))),aa_nat_int(F,I2)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
         => ? [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_3751_power__mono,axiom,
    ! [A2: real,B3: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),N))) ) ) ).

% power_mono
tff(fact_3752_power__mono,axiom,
    ! [A2: rat,B3: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B3),N))) ) ) ).

% power_mono
tff(fact_3753_power__mono,axiom,
    ! [A2: nat,B3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),N))) ) ) ).

% power_mono
tff(fact_3754_power__mono,axiom,
    ! [A2: int,B3: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(B3),N))) ) ) ).

% power_mono
tff(fact_3755_zero__le__power,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ).

% zero_le_power
tff(fact_3756_zero__le__power,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),N))) ) ).

% zero_le_power
tff(fact_3757_zero__le__power,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ).

% zero_le_power
tff(fact_3758_zero__le__power,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),N))) ) ).

% zero_le_power
tff(fact_3759_zero__less__power,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ).

% zero_less_power
tff(fact_3760_zero__less__power,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),N))) ) ).

% zero_less_power
tff(fact_3761_zero__less__power,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ).

% zero_less_power
tff(fact_3762_zero__less__power,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A2),N))) ) ).

% zero_less_power
tff(fact_3763_one__le__power,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ).

% one_le_power
tff(fact_3764_one__le__power,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A2),N))) ) ).

% one_le_power
tff(fact_3765_one__le__power,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ).

% one_le_power
tff(fact_3766_one__le__power,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A2),N))) ) ).

% one_le_power
tff(fact_3767_power__0,axiom,
    ! [A2: rat] : aa_nat_rat(power_power_rat(A2),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_3768_power__0,axiom,
    ! [A2: int] : aa_nat_int(power_power_int(A2),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_3769_power__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),zero_zero_nat) = one_one_nat ).

% power_0
tff(fact_3770_power__0,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_3771_power__0,axiom,
    ! [A2: complex] : aa_nat_complex(power_power_complex(A2),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_3772_power__less__imp__less__base,axiom,
    ! [A2: real,N: nat,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) ) ) ).

% power_less_imp_less_base
tff(fact_3773_power__less__imp__less__base,axiom,
    ! [A2: rat,N: nat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B3),N)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) ) ) ).

% power_less_imp_less_base
tff(fact_3774_power__less__imp__less__base,axiom,
    ! [A2: nat,N: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3)) ) ) ).

% power_less_imp_less_base
tff(fact_3775_power__less__imp__less__base,axiom,
    ! [A2: int,N: nat,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(B3),N)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3)) ) ) ).

% power_less_imp_less_base
tff(fact_3776_power__le__one,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),one_one_real)) ) ) ).

% power_le_one
tff(fact_3777_power__le__one,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),one_one_rat)) ) ) ).

% power_le_one
tff(fact_3778_power__le__one,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),one_one_nat)) ) ) ).

% power_le_one
tff(fact_3779_power__le__one,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A2),N)),one_one_int)) ) ) ).

% power_le_one
tff(fact_3780_power__inject__base,axiom,
    ! [A2: real,N: nat,B3: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),aa_nat_nat(suc,N)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3))
         => ( A2 = B3 ) ) ) ) ).

% power_inject_base
tff(fact_3781_power__inject__base,axiom,
    ! [A2: rat,N: nat,B3: rat] :
      ( ( aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N)) = aa_nat_rat(power_power_rat(B3),aa_nat_nat(suc,N)) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3))
         => ( A2 = B3 ) ) ) ) ).

% power_inject_base
tff(fact_3782_power__inject__base,axiom,
    ! [A2: nat,N: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),aa_nat_nat(suc,N)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B3))
         => ( A2 = B3 ) ) ) ) ).

% power_inject_base
tff(fact_3783_power__inject__base,axiom,
    ! [A2: int,N: nat,B3: int] :
      ( ( aa_nat_int(power_power_int(A2),aa_nat_nat(suc,N)) = aa_nat_int(power_power_int(B3),aa_nat_nat(suc,N)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3))
         => ( A2 = B3 ) ) ) ) ).

% power_inject_base
tff(fact_3784_power__le__imp__le__base,axiom,
    ! [A2: real,N: nat,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),aa_nat_nat(suc,N))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ) ).

% power_le_imp_le_base
tff(fact_3785_power__le__imp__le__base,axiom,
    ! [A2: rat,N: nat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N))),aa_nat_rat(power_power_rat(B3),aa_nat_nat(suc,N))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ) ).

% power_le_imp_le_base
tff(fact_3786_power__le__imp__le__base,axiom,
    ! [A2: nat,N: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3)) ) ) ).

% power_le_imp_le_base
tff(fact_3787_power__le__imp__le__base,axiom,
    ! [A2: int,N: nat,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A2),aa_nat_nat(suc,N))),aa_nat_int(power_power_int(B3),aa_nat_nat(suc,N))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ) ).

% power_le_imp_le_base
tff(fact_3788_power__gt1,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_3789_power__gt1,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_3790_power__gt1,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_3791_power__gt1,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A2),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_3792_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ) ).

% power_0_left
tff(fact_3793_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ) ).

% power_0_left
tff(fact_3794_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).

% power_0_left
tff(fact_3795_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = zero_zero_real ) ) ) ).

% power_0_left
tff(fact_3796_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ) ).

% power_0_left
tff(fact_3797_power__less__imp__less__exp,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_3798_power__less__imp__less__exp,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),M)),aa_nat_rat(power_power_rat(A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_3799_power__less__imp__less__exp,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_3800_power__less__imp__less__exp,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A2),M)),aa_nat_int(power_power_int(A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_3801_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N4))) ) ) ).

% power_strict_increasing
tff(fact_3802_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(A2),N4))) ) ) ).

% power_strict_increasing
tff(fact_3803_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N4))) ) ) ).

% power_strict_increasing
tff(fact_3804_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(A2),N4))) ) ) ).

% power_strict_increasing
tff(fact_3805_zero__le__power__abs,axiom,
    ! [A2: code_integer,N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A2)),N))) ).

% zero_le_power_abs
tff(fact_3806_zero__le__power__abs,axiom,
    ! [A2: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),N))) ).

% zero_le_power_abs
tff(fact_3807_zero__le__power__abs,axiom,
    ! [A2: rat,N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A2)),N))) ).

% zero_le_power_abs
tff(fact_3808_zero__le__power__abs,axiom,
    ! [A2: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A2)),N))) ).

% zero_le_power_abs
tff(fact_3809_power__increasing,axiom,
    ! [N: nat,N4: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N4))) ) ) ).

% power_increasing
tff(fact_3810_power__increasing,axiom,
    ! [N: nat,N4: nat,A2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(A2),N4))) ) ) ).

% power_increasing
tff(fact_3811_power__increasing,axiom,
    ! [N: nat,N4: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N4))) ) ) ).

% power_increasing
tff(fact_3812_power__increasing,axiom,
    ! [N: nat,N4: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(A2),N4))) ) ) ).

% power_increasing
tff(fact_3813_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ).

% zero_power
tff(fact_3814_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ).

% zero_power
tff(fact_3815_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ).

% zero_power
tff(fact_3816_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = zero_zero_real ) ) ).

% zero_power
tff(fact_3817_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ).

% zero_power
tff(fact_3818_power__Suc__le__self,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N))),A2)) ) ) ).

% power_Suc_le_self
tff(fact_3819_power__Suc__le__self,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N))),A2)) ) ) ).

% power_Suc_le_self
tff(fact_3820_power__Suc__le__self,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N))),A2)) ) ) ).

% power_Suc_le_self
tff(fact_3821_power__Suc__le__self,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A2),aa_nat_nat(suc,N))),A2)) ) ) ).

% power_Suc_le_self
tff(fact_3822_power__Suc__less__one,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N))),one_one_real)) ) ) ).

% power_Suc_less_one
tff(fact_3823_power__Suc__less__one,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N))),one_one_rat)) ) ) ).

% power_Suc_less_one
tff(fact_3824_power__Suc__less__one,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N))),one_one_nat)) ) ) ).

% power_Suc_less_one
tff(fact_3825_power__Suc__less__one,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A2),aa_nat_nat(suc,N))),one_one_int)) ) ) ).

% power_Suc_less_one
tff(fact_3826_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_3827_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),N4)),aa_nat_rat(power_power_rat(A2),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_3828_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_3829_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A2),N4)),aa_nat_int(power_power_int(A2),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_3830_power__decreasing,axiom,
    ! [N: nat,N4: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ) ) ).

% power_decreasing
tff(fact_3831_power__decreasing,axiom,
    ! [N: nat,N4: nat,A2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N4)),aa_nat_rat(power_power_rat(A2),N))) ) ) ) ).

% power_decreasing
tff(fact_3832_power__decreasing,axiom,
    ! [N: nat,N4: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ) ) ).

% power_decreasing
tff(fact_3833_power__decreasing,axiom,
    ! [N: nat,N4: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A2),N4)),aa_nat_int(power_power_int(A2),N))) ) ) ) ).

% power_decreasing
tff(fact_3834_power__le__imp__le__exp,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_3835_power__le__imp__le__exp,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),M)),aa_nat_rat(power_power_rat(A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_3836_power__le__imp__le__exp,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_3837_power__le__imp__le__exp,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A2),M)),aa_nat_int(power_power_int(A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_3838_power__eq__imp__eq__base,axiom,
    ! [A2: real,N: nat,B3: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),N) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A2 = B3 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3839_power__eq__imp__eq__base,axiom,
    ! [A2: rat,N: nat,B3: rat] :
      ( ( aa_nat_rat(power_power_rat(A2),N) = aa_nat_rat(power_power_rat(B3),N) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A2 = B3 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3840_power__eq__imp__eq__base,axiom,
    ! [A2: nat,N: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A2 = B3 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3841_power__eq__imp__eq__base,axiom,
    ! [A2: int,N: nat,B3: int] :
      ( ( aa_nat_int(power_power_int(A2),N) = aa_nat_int(power_power_int(B3),N) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A2 = B3 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3842_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: real,B3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3))
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),N) )
          <=> ( A2 = B3 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3843_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: rat,B3: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3))
         => ( ( aa_nat_rat(power_power_rat(A2),N) = aa_nat_rat(power_power_rat(B3),N) )
          <=> ( A2 = B3 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3844_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B3))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),N) )
          <=> ( A2 = B3 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3845_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: int,B3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3))
         => ( ( aa_nat_int(power_power_int(A2),N) = aa_nat_int(power_power_int(B3),N) )
          <=> ( A2 = B3 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3846_self__le__power,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ) ).

% self_le_power
tff(fact_3847_self__le__power,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_nat_rat(power_power_rat(A2),N))) ) ) ).

% self_le_power
tff(fact_3848_self__le__power,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ) ).

% self_le_power
tff(fact_3849_self__le__power,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(power_power_int(A2),N))) ) ) ).

% self_le_power
tff(fact_3850_one__less__power,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ) ).

% one_less_power
tff(fact_3851_one__less__power,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A2),N))) ) ) ).

% one_less_power
tff(fact_3852_one__less__power,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ) ).

% one_less_power
tff(fact_3853_one__less__power,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A2),N))) ) ) ).

% one_less_power
tff(fact_3854_power__diff,axiom,
    ! [A2: rat,N: nat,M: nat] :
      ( ( A2 != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(power_power_rat(A2),M)),aa_nat_rat(power_power_rat(A2),N)) ) ) ) ).

% power_diff
tff(fact_3855_power__diff,axiom,
    ! [A2: int,N: nat,M: nat] :
      ( ( A2 != zero_zero_int )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A2),M)),aa_nat_int(power_power_int(A2),N)) ) ) ) ).

% power_diff
tff(fact_3856_power__diff,axiom,
    ! [A2: nat,N: nat,M: nat] :
      ( ( A2 != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ) ) ) ).

% power_diff
tff(fact_3857_power__diff,axiom,
    ! [A2: real,N: nat,M: nat] :
      ( ( A2 != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ) ) ) ).

% power_diff
tff(fact_3858_power__diff,axiom,
    ! [A2: complex,N: nat,M: nat] :
      ( ( A2 != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_complex(power_power_complex(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A2),M)),aa_nat_complex(power_power_complex(A2),N)) ) ) ) ).

% power_diff
tff(fact_3859_power__strict__mono,axiom,
    ! [A2: real,B3: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),N))) ) ) ) ).

% power_strict_mono
tff(fact_3860_power__strict__mono,axiom,
    ! [A2: rat,B3: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B3),N))) ) ) ) ).

% power_strict_mono
tff(fact_3861_power__strict__mono,axiom,
    ! [A2: nat,B3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),N))) ) ) ) ).

% power_strict_mono
tff(fact_3862_power__strict__mono,axiom,
    ! [A2: int,B3: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(B3),N))) ) ) ) ).

% power_strict_mono
tff(fact_3863_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),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,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_3864_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),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,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_3865_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_3866_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_3867_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),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,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_3868_add__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(X)),some_nat(Y)) = some_nat(Z) ) ) ).

% add_shift
tff(fact_3869_lemma__interval,axiom,
    ! [A2: real,X: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B3))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [Y4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y4))),D6))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),B3)) ) ) ) ) ) ).

% lemma_interval
tff(fact_3870_realpow__pos__nth__unique,axiom,
    ! [N: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
       => ? [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X5))
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X5),N) = A2 )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4))
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y4),N) = A2 ) )
               => ( Y4 = X5 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_3871_realpow__pos__nth,axiom,
    ! [N: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),N) = A2 ) ) ) ) ).

% realpow_pos_nth
tff(fact_3872_lemma__interval__lt,axiom,
    ! [A2: real,X: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B3))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [Y4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y4))),D6))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),B3)) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_3873_realpow__pos__nth2,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ? [R3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),aa_nat_nat(suc,N)) = A2 ) ) ) ).

% realpow_pos_nth2
tff(fact_3874_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_3875_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),size_size_list_int(Xs)))
     => ( nth_Pr3440142176431000676at_int(enumerate_int(N,Xs),M) = product_Pair_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M),aa_nat_int(nth_int(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_3876_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_VEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),size_s6755466524823107622T_VEBT(Xs)))
     => ( nth_Pr744662078594809490T_VEBT(enumerate_VEBT_VEBT(N,Xs),M) = produc599794634098209291T_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M),nth_VEBT_VEBT(Xs,M)) ) ) ).

% nth_enumerate_eq
tff(fact_3877_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_bool,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),size_size_list_bool(Xs)))
     => ( nth_Pr396318242284072693t_bool(enumerate_bool(N,Xs),M) = produc3913204875133826030t_bool(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M),aa_nat_bool(nth_bool(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_3878_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),size_size_list_nat(Xs)))
     => ( nth_Pr7617993195940197384at_nat(enumerate_nat(N,Xs),M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),aa_nat_nat(nth_nat(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_3879_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z)) ) ).

% zle_add1_eq_le
tff(fact_3880_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,X)),Y)) ) ).

% real_0_less_add_iff
tff(fact_3881_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(uminus_uminus_real,X))) ) ).

% real_add_less_0_iff
tff(fact_3882_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,X))) ) ).

% real_add_le_0_iff
tff(fact_3883_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,X)),Y)) ) ).

% real_0_le_add_iff
tff(fact_3884_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I2))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_ge_induct
tff(fact_3885_zless__add1__eq,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z))
        | ( W2 = Z ) ) ) ).

% zless_add1_eq
tff(fact_3886_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I))
     => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I2))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_gr_induct
tff(fact_3887_zle__iff__zadd,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z))
    <=> ? [N3: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,N3)) ) ).

% zle_iff_zadd
tff(fact_3888_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z))
    <=> ? [N3: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ).

% zless_iff_Suc_zadd
tff(fact_3889_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% odd_less_0_iff
tff(fact_3890_add1__zle__eq,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ).

% add1_zle_eq
tff(fact_3891_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z)) ) ).

% zless_imp_add1_zle
tff(fact_3892_int__induct,axiom,
    ! [P: fun_int_bool,K: int,I: int] :
      ( pp(aa_int_bool(P,K))
     => ( ! [I2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I2))
           => ( pp(aa_int_bool(P,I2))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_induct
tff(fact_3893_nat__less__real__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% nat_less_real_le
tff(fact_3894_nat__le__real__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real))) ) ).

% nat_le_real_less
tff(fact_3895_le__imp__0__less,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z))) ) ).

% le_imp_0_less
tff(fact_3896_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_3897_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_3898_sin__bound__lemma,axiom,
    ! [X: real,Y: real,U: real,V: real] :
      ( ( X = Y )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,U)),V))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),U)),Y))),V)) ) ) ).

% sin_bound_lemma
tff(fact_3899_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,A2: nat,B3: nat] :
      ( ! [A: nat,B: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B),A)) )
     => ( ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),zero_zero_nat))
       => ( ! [A: nat,B: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),B3)) ) ) ) ).

% Euclid_induct
tff(fact_3900_add__0__iff,axiom,
    ! [B3: real,A2: real] :
      ( ( B3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),A2) )
    <=> ( A2 = zero_zero_real ) ) ).

% add_0_iff
tff(fact_3901_add__0__iff,axiom,
    ! [B3: rat,A2: rat] :
      ( ( B3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),A2) )
    <=> ( A2 = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_3902_add__0__iff,axiom,
    ! [B3: nat,A2: nat] :
      ( ( B3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),A2) )
    <=> ( A2 = zero_zero_nat ) ) ).

% add_0_iff
tff(fact_3903_add__0__iff,axiom,
    ! [B3: int,A2: int] :
      ( ( B3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),A2) )
    <=> ( A2 = zero_zero_int ) ) ).

% add_0_iff
tff(fact_3904_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_3905_frac__unique__iff,axiom,
    ! [X: real,A2: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = A2 )
    <=> ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A2)),ring_1_Ints_real))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),one_one_real)) ) ) ).

% frac_unique_iff
tff(fact_3906_frac__unique__iff,axiom,
    ! [X: rat,A2: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = A2 )
    <=> ( pp(aa_set_rat_bool(member_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),A2)),ring_1_Ints_rat))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),one_one_rat)) ) ) ).

% frac_unique_iff
tff(fact_3907_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Ys2)))
       => ( nth_Pr4439495888332055232nt_int(zip_int_int(Xs,Ys2),I) = product_Pair_int_int(aa_nat_int(nth_int(Xs),I),aa_nat_int(nth_int(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_3908_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ys2)))
       => ( nth_Pr3474266648193625910T_VEBT(zip_int_VEBT_VEBT(Xs,Ys2),I) = produc3329399203697025711T_VEBT(aa_nat_int(nth_int(Xs),I),nth_VEBT_VEBT(Ys2,I)) ) ) ) ).

% nth_zip
tff(fact_3909_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Ys2)))
       => ( nth_Pr6495399312670224281t_bool(zip_int_bool(Xs,Ys2),I) = produc788913908665201810t_bool(aa_nat_int(nth_int(Xs),I),aa_nat_bool(nth_bool(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_3910_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Ys2)))
       => ( nth_Pr8617346907841251940nt_nat(zip_int_nat(Xs,Ys2),I) = product_Pair_int_nat(aa_nat_int(nth_int(Xs),I),aa_nat_nat(nth_nat(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_3911_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Ys2)))
       => ( nth_Pr6837108013167703752BT_int(zip_VEBT_VEBT_int(Xs,Ys2),I) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,I),aa_nat_int(nth_int(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_3912_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ys2)))
       => ( nth_Pr4953567300277697838T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys2),I) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,I),nth_VEBT_VEBT(Ys2,I)) ) ) ) ).

% nth_zip
tff(fact_3913_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Ys2)))
       => ( nth_Pr1289831921885633937T_bool(zip_VEBT_VEBT_bool(Xs,Ys2),I) = produc3035007008238225530T_bool(nth_VEBT_VEBT(Xs,I),aa_nat_bool(nth_bool(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_3914_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Ys2)))
       => ( nth_Pr1791586995822124652BT_nat(zip_VEBT_VEBT_nat(Xs,Ys2),I) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,I),aa_nat_nat(nth_nat(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_3915_nth__zip,axiom,
    ! [I: nat,Xs: list_bool,Ys2: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Ys2)))
       => ( nth_Pr7231943226834047147ol_int(zip_bool_int(Xs,Ys2),I) = produc8871513391732212460ol_int(aa_nat_bool(nth_bool(Xs),I),aa_nat_int(nth_int(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_3916_nth__zip,axiom,
    ! [I: nat,Xs: list_bool,Ys2: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ys2)))
       => ( nth_Pr3250499561091541003T_VEBT(zip_bool_VEBT_VEBT(Xs,Ys2),I) = produc9074556814250467700T_VEBT(aa_nat_bool(nth_bool(Xs),I),nth_VEBT_VEBT(Ys2,I)) ) ) ) ).

% nth_zip
tff(fact_3917_exp__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y))) ) ).

% exp_less_mono
tff(fact_3918_exp__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% exp_less_cancel_iff
tff(fact_3919_exp__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% exp_le_cancel_iff
tff(fact_3920_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_3921_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_3922_frac__eq__0__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = zero_zero_real )
    <=> pp(aa_set_real_bool(member_real(X),ring_1_Ints_real)) ) ).

% frac_eq_0_iff
tff(fact_3923_frac__eq__0__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = zero_zero_rat )
    <=> pp(aa_set_rat_bool(member_rat(X),ring_1_Ints_rat)) ) ).

% frac_eq_0_iff
tff(fact_3924_exp__less__one__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% exp_less_one_iff
tff(fact_3925_one__less__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% one_less_exp_iff
tff(fact_3926_exp__le__one__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% exp_le_one_iff
tff(fact_3927_one__le__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(exp_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% one_le_exp_iff
tff(fact_3928_frac__gt__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,X)))
    <=> ~ pp(aa_set_real_bool(member_real(X),ring_1_Ints_real)) ) ).

% frac_gt_0_iff
tff(fact_3929_frac__gt__0__iff,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X)))
    <=> ~ pp(aa_set_rat_bool(member_rat(X),ring_1_Ints_rat)) ) ).

% frac_gt_0_iff
tff(fact_3930_exp__not__eq__zero,axiom,
    ! [X: real] : aa_real_real(exp_real,X) != zero_zero_real ).

% exp_not_eq_zero
tff(fact_3931_exp__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% exp_less_cancel
tff(fact_3932_Ints__0,axiom,
    pp(aa_set_real_bool(member_real(zero_zero_real),ring_1_Ints_real)) ).

% Ints_0
tff(fact_3933_Ints__0,axiom,
    pp(aa_set_rat_bool(member_rat(zero_zero_rat),ring_1_Ints_rat)) ).

% Ints_0
tff(fact_3934_Ints__0,axiom,
    pp(aa_set_int_bool(member_int(zero_zero_int),ring_1_Ints_int)) ).

% Ints_0
tff(fact_3935_not__exp__less__zero,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),zero_zero_real)) ).

% not_exp_less_zero
tff(fact_3936_exp__gt__zero,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(exp_real,X))) ).

% exp_gt_zero
tff(fact_3937_exp__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X5: real] : aa_real_real(exp_real,X5) = Y ) ).

% exp_total
tff(fact_3938_exp__ge__zero,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,X))) ).

% exp_ge_zero
tff(fact_3939_not__exp__le__zero,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),zero_zero_real)) ).

% not_exp_le_zero
tff(fact_3940_exp__ge__add__one__self,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(exp_real,X))) ).

% exp_ge_add_one_self
tff(fact_3941_exp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X))) ) ).

% exp_gt_one
tff(fact_3942_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(exp_real,X))) ) ).

% exp_ge_add_one_self_aux
tff(fact_3943_Ints__double__eq__0__iff,axiom,
    ! [A2: real] :
      ( pp(aa_set_real_bool(member_real(A2),ring_1_Ints_real))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2) = zero_zero_real )
      <=> ( A2 = zero_zero_real ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_3944_Ints__double__eq__0__iff,axiom,
    ! [A2: rat] :
      ( pp(aa_set_rat_bool(member_rat(A2),ring_1_Ints_rat))
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),A2) = zero_zero_rat )
      <=> ( A2 = zero_zero_rat ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_3945_Ints__double__eq__0__iff,axiom,
    ! [A2: int] :
      ( pp(aa_set_int_bool(member_int(A2),ring_1_Ints_int))
     => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2) = zero_zero_int )
      <=> ( A2 = zero_zero_int ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_3946_lemma__exp__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y))
     => ? [X5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X5))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),one_one_real)))
          & ( aa_real_real(exp_real,X5) = Y ) ) ) ).

% lemma_exp_total
tff(fact_3947_Ints__odd__nonzero,axiom,
    ! [A2: complex] :
      ( pp(aa_set_complex_bool(member_complex(A2),ring_1_Ints_complex))
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(one_one_complex),A2)),A2) != zero_zero_complex ) ) ).

% Ints_odd_nonzero
tff(fact_3948_Ints__odd__nonzero,axiom,
    ! [A2: real] :
      ( pp(aa_set_real_bool(member_real(A2),ring_1_Ints_real))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),A2)),A2) != zero_zero_real ) ) ).

% Ints_odd_nonzero
tff(fact_3949_Ints__odd__nonzero,axiom,
    ! [A2: rat] :
      ( pp(aa_set_rat_bool(member_rat(A2),ring_1_Ints_rat))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),A2)),A2) != zero_zero_rat ) ) ).

% Ints_odd_nonzero
tff(fact_3950_Ints__odd__nonzero,axiom,
    ! [A2: int] :
      ( pp(aa_set_int_bool(member_int(A2),ring_1_Ints_int))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A2)),A2) != zero_zero_int ) ) ).

% Ints_odd_nonzero
tff(fact_3951_exp__divide__power__eq,axiom,
    ! [N: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,aa_complex_complex(divide1717551699836669952omplex(X),aa_nat_complex(semiri8010041392384452111omplex,N)))),N) = aa_complex_complex(exp_complex,X) ) ) ).

% exp_divide_power_eq
tff(fact_3952_exp__divide__power__eq,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))),N) = aa_real_real(exp_real,X) ) ) ).

% exp_divide_power_eq
tff(fact_3953_Ints__odd__less__0,axiom,
    ! [A2: real] :
      ( pp(aa_set_real_bool(member_real(A2),ring_1_Ints_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),A2)),A2)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ) ).

% Ints_odd_less_0
tff(fact_3954_Ints__odd__less__0,axiom,
    ! [A2: rat] :
      ( pp(aa_set_rat_bool(member_rat(A2),ring_1_Ints_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),A2)),A2)),zero_zero_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ) ).

% Ints_odd_less_0
tff(fact_3955_Ints__odd__less__0,axiom,
    ! [A2: int] :
      ( pp(aa_set_int_bool(member_int(A2),ring_1_Ints_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A2)),A2)),zero_zero_int))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ) ).

% Ints_odd_less_0
tff(fact_3956_Ints__nonzero__abs__ge1,axiom,
    ! [X: code_integer] :
      ( member_Code_integer(X,ring_11222124179247155820nteger)
     => ( ( X != zero_z3403309356797280102nteger )
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),abs_abs_Code_integer(X))) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_3957_Ints__nonzero__abs__ge1,axiom,
    ! [X: real] :
      ( pp(aa_set_real_bool(member_real(X),ring_1_Ints_real))
     => ( ( X != zero_zero_real )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(abs_abs_real,X))) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_3958_Ints__nonzero__abs__ge1,axiom,
    ! [X: rat] :
      ( pp(aa_set_rat_bool(member_rat(X),ring_1_Ints_rat))
     => ( ( X != zero_zero_rat )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(abs_abs_rat,X))) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_3959_Ints__nonzero__abs__ge1,axiom,
    ! [X: int] :
      ( pp(aa_set_int_bool(member_int(X),ring_1_Ints_int))
     => ( ( X != zero_zero_int )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(abs_abs_int,X))) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_3960_Ints__nonzero__abs__less1,axiom,
    ! [X: code_integer] :
      ( member_Code_integer(X,ring_11222124179247155820nteger)
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(X)),one_one_Code_integer))
       => ( X = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_3961_Ints__nonzero__abs__less1,axiom,
    ! [X: real] :
      ( pp(aa_set_real_bool(member_real(X),ring_1_Ints_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
       => ( X = zero_zero_real ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_3962_Ints__nonzero__abs__less1,axiom,
    ! [X: rat] :
      ( pp(aa_set_rat_bool(member_rat(X),ring_1_Ints_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat))
       => ( X = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_3963_Ints__nonzero__abs__less1,axiom,
    ! [X: int] :
      ( pp(aa_set_int_bool(member_int(X),ring_1_Ints_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int))
       => ( X = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_3964_Ints__eq__abs__less1,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( member_Code_integer(X,ring_11222124179247155820nteger)
     => ( member_Code_integer(Y,ring_11222124179247155820nteger)
       => ( ( X = Y )
        <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,Y))),one_one_Code_integer)) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_3965_Ints__eq__abs__less1,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_set_real_bool(member_real(X),ring_1_Ints_real))
     => ( pp(aa_set_real_bool(member_real(Y),ring_1_Ints_real))
       => ( ( X = Y )
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))),one_one_real)) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_3966_Ints__eq__abs__less1,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_set_rat_bool(member_rat(X),ring_1_Ints_rat))
     => ( pp(aa_set_rat_bool(member_rat(Y),ring_1_Ints_rat))
       => ( ( X = Y )
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y))),one_one_rat)) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_3967_Ints__eq__abs__less1,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_set_int_bool(member_int(X),ring_1_Ints_int))
     => ( pp(aa_set_int_bool(member_int(Y),ring_1_Ints_int))
       => ( ( X = Y )
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y))),one_one_int)) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_3968_frac__neg,axiom,
    ! [X: real] :
      ( ( pp(aa_set_real_bool(member_real(X),ring_1_Ints_real))
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(uminus_uminus_real,X)) = zero_zero_real ) )
      & ( ~ pp(aa_set_real_bool(member_real(X),ring_1_Ints_real))
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(archim2898591450579166408c_real,X)) ) ) ) ).

% frac_neg
tff(fact_3969_frac__neg,axiom,
    ! [X: rat] :
      ( ( pp(aa_set_rat_bool(member_rat(X),ring_1_Ints_rat))
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(uminus_uminus_rat,X)) = zero_zero_rat ) )
      & ( ~ pp(aa_set_rat_bool(member_rat(X),ring_1_Ints_rat))
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(uminus_uminus_rat,X)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_rat_rat(archimedean_frac_rat,X)) ) ) ) ).

% frac_neg
tff(fact_3970_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,X))) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_3971_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_3972_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_3973_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_3974_artanh__minus__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,X)) ) ) ).

% artanh_minus_real
tff(fact_3975_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_3976_sinh__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sinh_real,X) = zero_zero_real )
    <=> pp(aa_set_real_bool(member_real(aa_real_real(exp_real,X)),insert_real(one_one_real,insert_real(aa_real_real(uminus_uminus_real,one_one_real),bot_bot_set_real)))) ) ).

% sinh_zero_iff
tff(fact_3977_sinh__zero__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(sinh_complex,X) = zero_zero_complex )
    <=> pp(aa_set_complex_bool(member_complex(aa_complex_complex(exp_complex,X)),insert_complex(one_one_complex,insert_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex),bot_bot_set_complex)))) ) ).

% sinh_zero_iff
tff(fact_3978_Gcd__remove0__nat,axiom,
    ! [M5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M5))
     => ( aa_set_nat_nat(gcd_Gcd_nat,M5) = aa_set_nat_nat(gcd_Gcd_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,M5),insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_3979_log__of__power__le,axiom,
    ! [M: nat,B3: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B3),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_le
tff(fact_3980_sinh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% sinh_real_less_iff
tff(fact_3981_sinh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% sinh_real_le_iff
tff(fact_3982_ln__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).

% ln_less_cancel_iff
tff(fact_3983_ln__inj__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( ( aa_real_real(ln_ln_real,X) = aa_real_real(ln_ln_real,Y) )
        <=> ( X = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_3984_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sinh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% sinh_real_pos_iff
tff(fact_3985_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% sinh_real_neg_iff
tff(fact_3986_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% sinh_real_nonpos_iff
tff(fact_3987_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% sinh_real_nonneg_iff
tff(fact_3988_sinh__0,axiom,
    aa_real_real(sinh_real,zero_zero_real) = zero_zero_real ).

% sinh_0
tff(fact_3989_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ).

% ln_le_cancel_iff
tff(fact_3990_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% ln_less_zero_iff
tff(fact_3991_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% ln_gt_zero_iff
tff(fact_3992_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( aa_real_real(ln_ln_real,X) = zero_zero_real )
      <=> ( X = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_3993_Gcd__empty,axiom,
    aa_set_nat_nat(gcd_Gcd_nat,bot_bot_set_nat) = zero_zero_nat ).

% Gcd_empty
tff(fact_3994_Gcd__empty,axiom,
    gcd_Gcd_int(bot_bot_set_int) = zero_zero_int ).

% Gcd_empty
tff(fact_3995_exp__ln,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X)) = X ) ) ).

% exp_ln
tff(fact_3996_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X)) = X )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% exp_ln_iff
tff(fact_3997_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% ln_ge_zero_iff
tff(fact_3998_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ).

% ln_le_zero_iff
tff(fact_3999_zero__less__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A2),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_4000_log__less__zero__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_4001_one__less__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A2),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X)) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_4002_log__less__one__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A2)) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_4003_log__less__cancel__iff,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A2),Y)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_4004_log__eq__one,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),A2) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_4005_Gcd__0__iff,axiom,
    ! [A3: set_nat] :
      ( ( aa_set_nat_nat(gcd_Gcd_nat,A3) = zero_zero_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ).

% Gcd_0_iff
tff(fact_4006_Gcd__0__iff,axiom,
    ! [A3: set_int] :
      ( ( gcd_Gcd_int(A3) = zero_zero_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(A3),insert_int(zero_zero_int,bot_bot_set_int))) ) ).

% Gcd_0_iff
tff(fact_4007_log__le__cancel__iff,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A2),Y)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_4008_log__le__one__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A2)) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_4009_one__le__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A2),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X)) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_4010_log__le__zero__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_4011_zero__le__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A2),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_4012_log__pow__cancel,axiom,
    ! [A2: real,B3: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),B3)) = aa_nat_real(semiri5074537144036343181t_real,B3) ) ) ) ).

% log_pow_cancel
tff(fact_4013_ln__less__self,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),X)) ) ).

% ln_less_self
tff(fact_4014_ln__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),X)) ) ).

% ln_bound
tff(fact_4015_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_4016_ln__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real)) ) ) ).

% ln_less_zero
tff(fact_4017_ln__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X))) ) ).

% ln_gt_zero
tff(fact_4018_ln__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X))) ) ).

% ln_ge_zero
tff(fact_4019_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_4020_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X)) ) ).

% ln_add_one_self_le_self
tff(fact_4021_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( aa_real_real(ln_ln_real,X) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real) )
       => ( X = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_4022_ln__div,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_div
tff(fact_4023_log__base__change,axiom,
    ! [A2: real,B3: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B3),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A2),B3)) ) ) ) ).

% log_base_change
tff(fact_4024_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,Y)),X)) ) ) ).

% ln_ge_iff
tff(fact_4025_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(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,X)),X))) ) ) ).

% ln_x_over_x_mono
tff(fact_4026_less__log__of__power,axiom,
    ! [B3: real,N: nat,M: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),N)),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B3),M))) ) ) ).

% less_log_of_power
tff(fact_4027_log__of__power__eq,axiom,
    ! [M: nat,B3: real,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),N) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
       => ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_rea7340605660300375871l_real(log,B3),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ) ).

% log_of_power_eq
tff(fact_4028_ln__le__minus__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real))) ) ).

% ln_le_minus_one
tff(fact_4029_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),Y))) ) ) ).

% ln_diff_le
tff(fact_4030_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X)) ) ).

% ln_add_one_self_le_self2
tff(fact_4031_log__divide,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A2),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_4032_le__log__of__power,axiom,
    ! [B3: real,N: nat,M: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),N)),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B3),M))) ) ) ).

% le_log_of_power
tff(fact_4033_log__base__pow,axiom,
    ! [A2: real,N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% log_base_pow
tff(fact_4034_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X))),aa_real_real(uminus_uminus_real,X))) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_4035_log__of__power__less,axiom,
    ! [M: nat,B3: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B3),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_less
tff(fact_4036_log__root,axiom,
    ! [N: nat,A2: real,B3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B3),aa_real_real(aa_nat_fun_real_real(root,N),A2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B3),A2)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% log_root
tff(fact_4037_decr__lemma,axiom,
    ! [D: int,X: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D))),Z)) ) ).

% decr_lemma
tff(fact_4038_incr__lemma,axiom,
    ! [D: int,Z: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D)))) ) ).

% incr_lemma
tff(fact_4039_Bernoulli__inequality,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),N))) ) ).

% Bernoulli_inequality
tff(fact_4040_linear__plus__1__le__power,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)),N))) ) ).

% linear_plus_1_le_power
tff(fact_4041_find__Some__iff,axiom,
    ! [P: fun_int_bool,Xs: list_int,X: int] :
      ( ( find_int(P,Xs) = some_int(X) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
          & pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I4)))
          & ( X = aa_nat_int(nth_int(Xs),I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),J3))) ) ) ) ).

% find_Some_iff
tff(fact_4042_find__Some__iff,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( ( find_P8199882355184865565at_nat(P,Xs) = some_P7363390416028606310at_nat(X) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s5460976970255530739at_nat(Xs)))
          & pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,I4)))
          & ( X = nth_Pr7617993195940197384at_nat(Xs,I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,J3))) ) ) ) ).

% find_Some_iff
tff(fact_4043_find__Some__iff,axiom,
    ! [P: fun_num_bool,Xs: list_num,X: num] :
      ( ( find_num(P,Xs) = some_num(X) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_num(Xs)))
          & pp(aa_num_bool(P,aa_nat_num(nth_num(Xs),I4)))
          & ( X = aa_nat_num(nth_num(Xs),I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_num_bool(P,aa_nat_num(nth_num(Xs),J3))) ) ) ) ).

% find_Some_iff
tff(fact_4044_find__Some__iff,axiom,
    ! [P: fun_VEBT_VEBT_bool,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( find_VEBT_VEBT(P,Xs) = some_VEBT_VEBT(X) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs)))
          & pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I4)))
          & ( X = nth_VEBT_VEBT(Xs,I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,J3))) ) ) ) ).

% find_Some_iff
tff(fact_4045_find__Some__iff,axiom,
    ! [P: fun_bool_bool,Xs: list_bool,X: bool] :
      ( ( find_bool(P,Xs) = some_bool(X) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Xs)))
          & pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I4)))
          & ( pp(X)
          <=> pp(aa_nat_bool(nth_bool(Xs),I4)) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),J3))) ) ) ) ).

% find_Some_iff
tff(fact_4046_find__Some__iff,axiom,
    ! [P: fun_nat_bool,Xs: list_nat,X: nat] :
      ( ( find_nat(P,Xs) = some_nat(X) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
          & pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I4)))
          & ( X = aa_nat_nat(nth_nat(Xs),I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),J3))) ) ) ) ).

% find_Some_iff
tff(fact_4047_find__Some__iff2,axiom,
    ! [X: int,P: fun_int_bool,Xs: list_int] :
      ( ( some_int(X) = find_int(P,Xs) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
          & pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I4)))
          & ( X = aa_nat_int(nth_int(Xs),I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),J3))) ) ) ) ).

% find_Some_iff2
tff(fact_4048_find__Some__iff2,axiom,
    ! [X: product_prod_nat_nat,P: fun_Pr7757163693017731481t_bool,Xs: list_P6011104703257516679at_nat] :
      ( ( some_P7363390416028606310at_nat(X) = find_P8199882355184865565at_nat(P,Xs) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s5460976970255530739at_nat(Xs)))
          & pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,I4)))
          & ( X = nth_Pr7617993195940197384at_nat(Xs,I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,J3))) ) ) ) ).

% find_Some_iff2
tff(fact_4049_find__Some__iff2,axiom,
    ! [X: num,P: fun_num_bool,Xs: list_num] :
      ( ( some_num(X) = find_num(P,Xs) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_num(Xs)))
          & pp(aa_num_bool(P,aa_nat_num(nth_num(Xs),I4)))
          & ( X = aa_nat_num(nth_num(Xs),I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_num_bool(P,aa_nat_num(nth_num(Xs),J3))) ) ) ) ).

% find_Some_iff2
tff(fact_4050_find__Some__iff2,axiom,
    ! [X: vEBT_VEBT,P: fun_VEBT_VEBT_bool,Xs: list_VEBT_VEBT] :
      ( ( some_VEBT_VEBT(X) = find_VEBT_VEBT(P,Xs) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs)))
          & pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I4)))
          & ( X = nth_VEBT_VEBT(Xs,I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,J3))) ) ) ) ).

% find_Some_iff2
tff(fact_4051_find__Some__iff2,axiom,
    ! [X: bool,P: fun_bool_bool,Xs: list_bool] :
      ( ( some_bool(X) = find_bool(P,Xs) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Xs)))
          & pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I4)))
          & ( pp(X)
          <=> pp(aa_nat_bool(nth_bool(Xs),I4)) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),J3))) ) ) ) ).

% find_Some_iff2
tff(fact_4052_find__Some__iff2,axiom,
    ! [X: nat,P: fun_nat_bool,Xs: list_nat] :
      ( ( some_nat(X) = find_nat(P,Xs) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
          & pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I4)))
          & ( X = aa_nat_nat(nth_nat(Xs),I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),J3))) ) ) ) ).

% find_Some_iff2
tff(fact_4053_mult__cancel__right,axiom,
    ! [A2: complex,C: complex,B3: complex] :
      ( ( aa_complex_complex(times_times_complex(A2),C) = aa_complex_complex(times_times_complex(B3),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_right
tff(fact_4054_mult__cancel__right,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C) )
    <=> ( ( C = zero_zero_real )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_right
tff(fact_4055_mult__cancel__right,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_right
tff(fact_4056_mult__cancel__right,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_right
tff(fact_4057_mult__cancel__right,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B3),C) )
    <=> ( ( C = zero_zero_int )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_right
tff(fact_4058_mult__cancel__left,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A2) = aa_complex_complex(times_times_complex(C),B3) )
    <=> ( ( C = zero_zero_complex )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_left
tff(fact_4059_mult__cancel__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3) )
    <=> ( ( C = zero_zero_real )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_left
tff(fact_4060_mult__cancel__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3) )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_left
tff(fact_4061_mult__cancel__left,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3) )
    <=> ( ( C = zero_zero_nat )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_left
tff(fact_4062_mult__cancel__left,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B3) )
    <=> ( ( C = zero_zero_int )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_left
tff(fact_4063_mult__eq__0__iff,axiom,
    ! [A2: complex,B3: complex] :
      ( ( aa_complex_complex(times_times_complex(A2),B3) = zero_zero_complex )
    <=> ( ( A2 = zero_zero_complex )
        | ( B3 = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
tff(fact_4064_mult__eq__0__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3) = zero_zero_real )
    <=> ( ( A2 = zero_zero_real )
        | ( B3 = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_4065_mult__eq__0__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3) = zero_zero_rat )
    <=> ( ( A2 = zero_zero_rat )
        | ( B3 = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_4066_mult__eq__0__iff,axiom,
    ! [A2: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B3) = zero_zero_nat )
    <=> ( ( A2 = zero_zero_nat )
        | ( B3 = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_4067_mult__eq__0__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3) = zero_zero_int )
    <=> ( ( A2 = zero_zero_int )
        | ( B3 = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_4068_mult__zero__right,axiom,
    ! [A2: complex] : aa_complex_complex(times_times_complex(A2),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_4069_mult__zero__right,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_4070_mult__zero__right,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_4071_mult__zero__right,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),zero_zero_nat) = zero_zero_nat ).

% mult_zero_right
tff(fact_4072_mult__zero__right,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_4073_mult__zero__left,axiom,
    ! [A2: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A2) = zero_zero_complex ).

% mult_zero_left
tff(fact_4074_mult__zero__left,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_zero_real),A2) = zero_zero_real ).

% mult_zero_left
tff(fact_4075_mult__zero__left,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A2) = zero_zero_rat ).

% mult_zero_left
tff(fact_4076_mult__zero__left,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A2) = zero_zero_nat ).

% mult_zero_left
tff(fact_4077_mult__zero__left,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),A2) = zero_zero_int ).

% mult_zero_left
tff(fact_4078_mult_Oright__neutral,axiom,
    ! [A2: complex] : aa_complex_complex(times_times_complex(A2),one_one_complex) = A2 ).

% mult.right_neutral
tff(fact_4079_mult_Oright__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),one_one_real) = A2 ).

% mult.right_neutral
tff(fact_4080_mult_Oright__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),one_one_rat) = A2 ).

% mult.right_neutral
tff(fact_4081_mult_Oright__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),one_one_nat) = A2 ).

% mult.right_neutral
tff(fact_4082_mult_Oright__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),one_one_int) = A2 ).

% mult.right_neutral
tff(fact_4083_mult__1,axiom,
    ! [A2: complex] : aa_complex_complex(times_times_complex(one_one_complex),A2) = A2 ).

% mult_1
tff(fact_4084_mult__1,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A2) = A2 ).

% mult_1
tff(fact_4085_mult__1,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A2) = A2 ).

% mult_1
tff(fact_4086_mult__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A2) = A2 ).

% mult_1
tff(fact_4087_mult__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A2) = A2 ).

% mult_1
tff(fact_4088_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,N)) ).

% of_nat_mult
tff(fact_4089_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_mult
tff(fact_4090_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_mult
tff(fact_4091_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_mult
tff(fact_4092_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_mult
tff(fact_4093_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_4094_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_4095_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_4096_mult__cancel__right2,axiom,
    ! [A2: complex,C: complex] :
      ( ( aa_complex_complex(times_times_complex(A2),C) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A2 = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_4097_mult__cancel__right2,axiom,
    ! [A2: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( A2 = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_4098_mult__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_4099_mult__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_4100_mult__cancel__right1,axiom,
    ! [C: complex,B3: complex] :
      ( ( C = aa_complex_complex(times_times_complex(B3),C) )
    <=> ( ( C = zero_zero_complex )
        | ( B3 = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_4101_mult__cancel__right1,axiom,
    ! [C: real,B3: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C) )
    <=> ( ( C = zero_zero_real )
        | ( B3 = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_4102_mult__cancel__right1,axiom,
    ! [C: rat,B3: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B3 = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_4103_mult__cancel__right1,axiom,
    ! [C: int,B3: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B3),C) )
    <=> ( ( C = zero_zero_int )
        | ( B3 = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_4104_mult__cancel__left2,axiom,
    ! [C: complex,A2: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A2) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A2 = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_4105_mult__cancel__left2,axiom,
    ! [C: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2) = C )
    <=> ( ( C = zero_zero_real )
        | ( A2 = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_4106_mult__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_4107_mult__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A2) = C )
    <=> ( ( C = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_4108_mult__cancel__left1,axiom,
    ! [C: complex,B3: complex] :
      ( ( C = aa_complex_complex(times_times_complex(C),B3) )
    <=> ( ( C = zero_zero_complex )
        | ( B3 = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_4109_mult__cancel__left1,axiom,
    ! [C: real,B3: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3) )
    <=> ( ( C = zero_zero_real )
        | ( B3 = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_4110_mult__cancel__left1,axiom,
    ! [C: rat,B3: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3) )
    <=> ( ( C = zero_zero_rat )
        | ( B3 = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_4111_mult__cancel__left1,axiom,
    ! [C: int,B3: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B3) )
    <=> ( ( C = zero_zero_int )
        | ( B3 = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_4112_div__mult__mult1,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3) ) ) ).

% div_mult_mult1
tff(fact_4113_div__mult__mult1,axiom,
    ! [C: nat,A2: nat,B3: 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),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B3) ) ) ).

% div_mult_mult1
tff(fact_4114_div__mult__mult2,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3) ) ) ).

% div_mult_mult2
tff(fact_4115_div__mult__mult2,axiom,
    ! [C: nat,A2: nat,B3: 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,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B3) ) ) ).

% div_mult_mult2
tff(fact_4116_div__mult__mult1__if,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ( C = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3)) = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3) ) ) ) ).

% div_mult_mult1_if
tff(fact_4117_div__mult__mult1__if,axiom,
    ! [C: nat,A2: nat,B3: 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),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3)) = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B3) ) ) ) ).

% div_mult_mult1_if
tff(fact_4118_nonzero__mult__div__cancel__left,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)),A2) = B3 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_4119_nonzero__mult__div__cancel__left,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3)),A2) = B3 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_4120_nonzero__mult__div__cancel__left,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 != 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,A2),B3)),A2) = B3 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_4121_nonzero__mult__div__cancel__left,axiom,
    ! [A2: real,B3: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)),A2) = B3 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_4122_nonzero__mult__div__cancel__left,axiom,
    ! [A2: complex,B3: complex] :
      ( ( A2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A2),B3)),A2) = B3 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_4123_nonzero__mult__div__cancel__right,axiom,
    ! [B3: rat,A2: rat] :
      ( ( B3 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)),B3) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_4124_nonzero__mult__div__cancel__right,axiom,
    ! [B3: int,A2: int] :
      ( ( B3 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3)),B3) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_4125_nonzero__mult__div__cancel__right,axiom,
    ! [B3: nat,A2: nat] :
      ( ( B3 != 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,A2),B3)),B3) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_4126_nonzero__mult__div__cancel__right,axiom,
    ! [B3: real,A2: real] :
      ( ( B3 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)),B3) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_4127_nonzero__mult__div__cancel__right,axiom,
    ! [B3: complex,A2: complex] :
      ( ( B3 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A2),B3)),B3) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_4128_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ( C = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3)) = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_4129_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ( C = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3)) = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_4130_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( ( C = zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A2)),aa_complex_complex(times_times_complex(C),B3)) = zero_zero_complex ) )
      & ( ( C != zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A2)),aa_complex_complex(times_times_complex(C),B3)) = aa_complex_complex(divide1717551699836669952omplex(A2),B3) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_4131_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_4132_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_4133_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A2)),aa_complex_complex(times_times_complex(C),B3)) = aa_complex_complex(divide1717551699836669952omplex(A2),B3) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_4134_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_4135_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_4136_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A2)),aa_complex_complex(times_times_complex(B3),C)) = aa_complex_complex(divide1717551699836669952omplex(A2),B3) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_4137_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_4138_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_4139_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A2),C)),aa_complex_complex(times_times_complex(B3),C)) = aa_complex_complex(divide1717551699836669952omplex(A2),B3) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_4140_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_4141_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_4142_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A2),C)),aa_complex_complex(times_times_complex(C),B3)) = aa_complex_complex(divide1717551699836669952omplex(A2),B3) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_4143_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)))
    <=> ( X = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_4144_real__root__Suc__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),X) = X ).

% real_root_Suc_0
tff(fact_4145_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(aa_nat_fun_real_real(root,N),Y) )
      <=> ( X = Y ) ) ) ).

% real_root_eq_iff
tff(fact_4146_root__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),X) = zero_zero_real ).

% root_0
tff(fact_4147_div__mult__self1,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( B3 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3))),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)) ) ) ).

% div_mult_self1
tff(fact_4148_div__mult__self1,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( B3 != 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,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3))),B3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B3)) ) ) ).

% div_mult_self1
tff(fact_4149_div__mult__self2,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( B3 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C))),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)) ) ) ).

% div_mult_self2
tff(fact_4150_div__mult__self2,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( B3 != 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,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),C))),B3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B3)) ) ) ).

% div_mult_self2
tff(fact_4151_div__mult__self3,axiom,
    ! [B3: int,C: int,A2: int] :
      ( ( B3 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B3)),A2)),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)) ) ) ).

% div_mult_self3
tff(fact_4152_div__mult__self3,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( ( B3 != 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),B3)),A2)),B3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B3)) ) ) ).

% div_mult_self3
tff(fact_4153_div__mult__self4,axiom,
    ! [B3: int,C: int,A2: int] :
      ( ( B3 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),C)),A2)),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)) ) ) ).

% div_mult_self4
tff(fact_4154_div__mult__self4,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( ( B3 != 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,B3),C)),A2)),B3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B3)) ) ) ).

% div_mult_self4
tff(fact_4155_nonzero__divide__mult__cancel__right,axiom,
    ! [B3: rat,A2: rat] :
      ( ( B3 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A2) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_4156_nonzero__divide__mult__cancel__right,axiom,
    ! [B3: real,A2: real] :
      ( ( B3 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_4157_nonzero__divide__mult__cancel__right,axiom,
    ! [B3: complex,A2: complex] :
      ( ( B3 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(B3),aa_complex_complex(times_times_complex(A2),B3)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A2) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_4158_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),B3) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_4159_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: real,B3: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),B3) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_4160_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: complex,B3: complex] :
      ( ( A2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A2),aa_complex_complex(times_times_complex(A2),B3)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),B3) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_4161_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = zero_zero_real )
      <=> ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_4162_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% real_root_less_iff
tff(fact_4163_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% real_root_le_iff
tff(fact_4164_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = one_one_real )
      <=> ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_4165_real__root__one,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_4166_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ) ).

% real_root_lt_0_iff
tff(fact_4167_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y)) ) ) ).

% real_root_gt_0_iff
tff(fact_4168_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ) ).

% real_root_le_0_iff
tff(fact_4169_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y)) ) ) ).

% real_root_ge_0_iff
tff(fact_4170_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% real_root_lt_1_iff
tff(fact_4171_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y)) ) ) ).

% real_root_gt_1_iff
tff(fact_4172_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ).

% real_root_le_1_iff
tff(fact_4173_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y)) ) ) ).

% real_root_ge_1_iff
tff(fact_4174_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_4175_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A2: complex,B3: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A2),B3)),C) = aa_complex_complex(times_times_complex(A2),aa_complex_complex(times_times_complex(B3),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_4176_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A2: real,B3: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_4177_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A2: rat,B3: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_4178_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A2: nat,B3: 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,A2),B3)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_4179_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A2: int,B3: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_4180_mult_Oassoc,axiom,
    ! [A2: complex,B3: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A2),B3)),C) = aa_complex_complex(times_times_complex(A2),aa_complex_complex(times_times_complex(B3),C)) ).

% mult.assoc
tff(fact_4181_mult_Oassoc,axiom,
    ! [A2: real,B3: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C)) ).

% mult.assoc
tff(fact_4182_mult_Oassoc,axiom,
    ! [A2: rat,B3: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C)) ).

% mult.assoc
tff(fact_4183_mult_Oassoc,axiom,
    ! [A2: nat,B3: 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,A2),B3)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),C)) ).

% mult.assoc
tff(fact_4184_mult_Oassoc,axiom,
    ! [A2: int,B3: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C)) ).

% mult.assoc
tff(fact_4185_mult_Ocommute,axiom,
    ! [A2: complex,B3: complex] : aa_complex_complex(times_times_complex(A2),B3) = aa_complex_complex(times_times_complex(B3),A2) ).

% mult.commute
tff(fact_4186_mult_Ocommute,axiom,
    ! [A2: real,B3: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),A2) ).

% mult.commute
tff(fact_4187_mult_Ocommute,axiom,
    ! [A2: rat,B3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),A2) ).

% mult.commute
tff(fact_4188_mult_Ocommute,axiom,
    ! [A2: nat,B3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),A2) ).

% mult.commute
tff(fact_4189_mult_Ocommute,axiom,
    ! [A2: int,B3: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3) = aa_int_int(aa_int_fun_int_int(times_times_int,B3),A2) ).

% mult.commute
tff(fact_4190_mult_Oleft__commute,axiom,
    ! [B3: complex,A2: complex,C: complex] : aa_complex_complex(times_times_complex(B3),aa_complex_complex(times_times_complex(A2),C)) = aa_complex_complex(times_times_complex(A2),aa_complex_complex(times_times_complex(B3),C)) ).

% mult.left_commute
tff(fact_4191_mult_Oleft__commute,axiom,
    ! [B3: real,A2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C)) ).

% mult.left_commute
tff(fact_4192_mult_Oleft__commute,axiom,
    ! [B3: rat,A2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C)) ).

% mult.left_commute
tff(fact_4193_mult_Oleft__commute,axiom,
    ! [B3: nat,A2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),C)) ).

% mult.left_commute
tff(fact_4194_mult_Oleft__commute,axiom,
    ! [B3: int,A2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B3),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C)) ).

% mult.left_commute
tff(fact_4195_mult__right__cancel,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A2),C) = aa_complex_complex(times_times_complex(B3),C) )
      <=> ( A2 = B3 ) ) ) ).

% mult_right_cancel
tff(fact_4196_mult__right__cancel,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C) )
      <=> ( A2 = B3 ) ) ) ).

% mult_right_cancel
tff(fact_4197_mult__right__cancel,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C) )
      <=> ( A2 = B3 ) ) ) ).

% mult_right_cancel
tff(fact_4198_mult__right__cancel,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),C) )
      <=> ( A2 = B3 ) ) ) ).

% mult_right_cancel
tff(fact_4199_mult__right__cancel,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B3),C) )
      <=> ( A2 = B3 ) ) ) ).

% mult_right_cancel
tff(fact_4200_mult__left__cancel,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(C),A2) = aa_complex_complex(times_times_complex(C),B3) )
      <=> ( A2 = B3 ) ) ) ).

% mult_left_cancel
tff(fact_4201_mult__left__cancel,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3) )
      <=> ( A2 = B3 ) ) ) ).

% mult_left_cancel
tff(fact_4202_mult__left__cancel,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3) )
      <=> ( A2 = B3 ) ) ) ).

% mult_left_cancel
tff(fact_4203_mult__left__cancel,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3) )
      <=> ( A2 = B3 ) ) ) ).

% mult_left_cancel
tff(fact_4204_mult__left__cancel,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B3) )
      <=> ( A2 = B3 ) ) ) ).

% mult_left_cancel
tff(fact_4205_no__zero__divisors,axiom,
    ! [A2: complex,B3: complex] :
      ( ( A2 != zero_zero_complex )
     => ( ( B3 != zero_zero_complex )
       => ( aa_complex_complex(times_times_complex(A2),B3) != zero_zero_complex ) ) ) ).

% no_zero_divisors
tff(fact_4206_no__zero__divisors,axiom,
    ! [A2: real,B3: real] :
      ( ( A2 != zero_zero_real )
     => ( ( B3 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_4207_no__zero__divisors,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( B3 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_4208_no__zero__divisors,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( B3 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B3) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_4209_no__zero__divisors,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 != zero_zero_int )
     => ( ( B3 != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_4210_divisors__zero,axiom,
    ! [A2: complex,B3: complex] :
      ( ( aa_complex_complex(times_times_complex(A2),B3) = zero_zero_complex )
     => ( ( A2 = zero_zero_complex )
        | ( B3 = zero_zero_complex ) ) ) ).

% divisors_zero
tff(fact_4211_divisors__zero,axiom,
    ! [A2: real,B3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3) = zero_zero_real )
     => ( ( A2 = zero_zero_real )
        | ( B3 = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_4212_divisors__zero,axiom,
    ! [A2: rat,B3: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3) = zero_zero_rat )
     => ( ( A2 = zero_zero_rat )
        | ( B3 = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_4213_divisors__zero,axiom,
    ! [A2: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B3) = zero_zero_nat )
     => ( ( A2 = zero_zero_nat )
        | ( B3 = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_4214_divisors__zero,axiom,
    ! [A2: int,B3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3) = zero_zero_int )
     => ( ( A2 = zero_zero_int )
        | ( B3 = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_4215_mult__not__zero,axiom,
    ! [A2: complex,B3: complex] :
      ( ( aa_complex_complex(times_times_complex(A2),B3) != zero_zero_complex )
     => ( ( A2 != zero_zero_complex )
        & ( B3 != zero_zero_complex ) ) ) ).

% mult_not_zero
tff(fact_4216_mult__not__zero,axiom,
    ! [A2: real,B3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3) != zero_zero_real )
     => ( ( A2 != zero_zero_real )
        & ( B3 != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_4217_mult__not__zero,axiom,
    ! [A2: rat,B3: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3) != zero_zero_rat )
     => ( ( A2 != zero_zero_rat )
        & ( B3 != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_4218_mult__not__zero,axiom,
    ! [A2: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B3) != zero_zero_nat )
     => ( ( A2 != zero_zero_nat )
        & ( B3 != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_4219_mult__not__zero,axiom,
    ! [A2: int,B3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3) != zero_zero_int )
     => ( ( A2 != zero_zero_int )
        & ( B3 != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_4220_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: complex] : aa_complex_complex(times_times_complex(one_one_complex),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_4221_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_4222_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_4223_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_4224_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_4225_mult_Ocomm__neutral,axiom,
    ! [A2: complex] : aa_complex_complex(times_times_complex(A2),one_one_complex) = A2 ).

% mult.comm_neutral
tff(fact_4226_mult_Ocomm__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),one_one_real) = A2 ).

% mult.comm_neutral
tff(fact_4227_mult_Ocomm__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),one_one_rat) = A2 ).

% mult.comm_neutral
tff(fact_4228_mult_Ocomm__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),one_one_nat) = A2 ).

% mult.comm_neutral
tff(fact_4229_mult_Ocomm__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),one_one_int) = A2 ).

% mult.comm_neutral
tff(fact_4230_mult__of__nat__commute,axiom,
    ! [X: nat,Y: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,X)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(semiri8010041392384452111omplex,X)) ).

% mult_of_nat_commute
tff(fact_4231_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(semiri681578069525770553at_rat,X)) ).

% mult_of_nat_commute
tff(fact_4232_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,X)) ).

% mult_of_nat_commute
tff(fact_4233_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,X)) ).

% mult_of_nat_commute
tff(fact_4234_mult__of__nat__commute,axiom,
    ! [X: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(semiri5074537144036343181t_real,X)) ).

% mult_of_nat_commute
tff(fact_4235_real__root__pos__pos__le,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ).

% real_root_pos_pos_le
tff(fact_4236_mult__mono,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),D))) ) ) ) ) ).

% mult_mono
tff(fact_4237_mult__mono,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),D))) ) ) ) ) ).

% mult_mono
tff(fact_4238_mult__mono,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),D))) ) ) ) ) ).

% mult_mono
tff(fact_4239_mult__mono,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),D))) ) ) ) ) ).

% mult_mono
tff(fact_4240_mult__mono_H,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),D))) ) ) ) ) ).

% mult_mono'
tff(fact_4241_mult__mono_H,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),D))) ) ) ) ) ).

% mult_mono'
tff(fact_4242_mult__mono_H,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),D))) ) ) ) ) ).

% mult_mono'
tff(fact_4243_mult__mono_H,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),D))) ) ) ) ) ).

% mult_mono'
tff(fact_4244_zero__le__square,axiom,
    ! [A2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),A2))) ).

% zero_le_square
tff(fact_4245_zero__le__square,axiom,
    ! [A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),A2))) ).

% zero_le_square
tff(fact_4246_zero__le__square,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),A2))) ).

% zero_le_square
tff(fact_4247_split__mult__pos__le,axiom,
    ! [A2: real,B3: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),zero_zero_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3))) ) ).

% split_mult_pos_le
tff(fact_4248_split__mult__pos__le,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),zero_zero_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3))) ) ).

% split_mult_pos_le
tff(fact_4249_split__mult__pos__le,axiom,
    ! [A2: int,B3: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),zero_zero_int)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3))) ) ).

% split_mult_pos_le
tff(fact_4250_mult__left__mono__neg,axiom,
    ! [B3: real,A2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3))) ) ) ).

% mult_left_mono_neg
tff(fact_4251_mult__left__mono__neg,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3))) ) ) ).

% mult_left_mono_neg
tff(fact_4252_mult__left__mono__neg,axiom,
    ! [B3: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3))) ) ) ).

% mult_left_mono_neg
tff(fact_4253_mult__nonpos__nonpos,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3))) ) ) ).

% mult_nonpos_nonpos
tff(fact_4254_mult__nonpos__nonpos,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3))) ) ) ).

% mult_nonpos_nonpos
tff(fact_4255_mult__nonpos__nonpos,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3))) ) ) ).

% mult_nonpos_nonpos
tff(fact_4256_mult__left__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3))) ) ) ).

% mult_left_mono
tff(fact_4257_mult__left__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3))) ) ) ).

% mult_left_mono
tff(fact_4258_mult__left__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3))) ) ) ).

% mult_left_mono
tff(fact_4259_mult__left__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3))) ) ) ).

% mult_left_mono
tff(fact_4260_mult__right__mono__neg,axiom,
    ! [B3: real,A2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C))) ) ) ).

% mult_right_mono_neg
tff(fact_4261_mult__right__mono__neg,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C))) ) ) ).

% mult_right_mono_neg
tff(fact_4262_mult__right__mono__neg,axiom,
    ! [B3: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C))) ) ) ).

% mult_right_mono_neg
tff(fact_4263_mult__right__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C))) ) ) ).

% mult_right_mono
tff(fact_4264_mult__right__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C))) ) ) ).

% mult_right_mono
tff(fact_4265_mult__right__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),C))) ) ) ).

% mult_right_mono
tff(fact_4266_mult__right__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C))) ) ) ).

% mult_right_mono
tff(fact_4267_mult__le__0__iff,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3)) ) ) ) ).

% mult_le_0_iff
tff(fact_4268_mult__le__0__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3)) ) ) ) ).

% mult_le_0_iff
tff(fact_4269_mult__le__0__iff,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3)) ) ) ) ).

% mult_le_0_iff
tff(fact_4270_split__mult__neg__le,axiom,
    ! [A2: real,B3: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)),zero_zero_real)) ) ).

% split_mult_neg_le
tff(fact_4271_split__mult__neg__le,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)),zero_zero_rat)) ) ).

% split_mult_neg_le
tff(fact_4272_split__mult__neg__le,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),zero_zero_nat)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B3)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B3)),zero_zero_nat)) ) ).

% split_mult_neg_le
tff(fact_4273_split__mult__neg__le,axiom,
    ! [A2: int,B3: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3)),zero_zero_int)) ) ).

% split_mult_neg_le
tff(fact_4274_mult__nonneg__nonneg,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3))) ) ) ).

% mult_nonneg_nonneg
tff(fact_4275_mult__nonneg__nonneg,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3))) ) ) ).

% mult_nonneg_nonneg
tff(fact_4276_mult__nonneg__nonneg,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B3))) ) ) ).

% mult_nonneg_nonneg
tff(fact_4277_mult__nonneg__nonneg,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3))) ) ) ).

% mult_nonneg_nonneg
tff(fact_4278_mult__nonneg__nonpos,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos
tff(fact_4279_mult__nonneg__nonpos,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_4280_mult__nonneg__nonpos,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B3)),zero_zero_nat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_4281_mult__nonneg__nonpos,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos
tff(fact_4282_mult__nonpos__nonneg,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)),zero_zero_real)) ) ) ).

% mult_nonpos_nonneg
tff(fact_4283_mult__nonpos__nonneg,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)),zero_zero_rat)) ) ) ).

% mult_nonpos_nonneg
tff(fact_4284_mult__nonpos__nonneg,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B3)),zero_zero_nat)) ) ) ).

% mult_nonpos_nonneg
tff(fact_4285_mult__nonpos__nonneg,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3)),zero_zero_int)) ) ) ).

% mult_nonpos_nonneg
tff(fact_4286_mult__nonneg__nonpos2,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),A2)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_4287_mult__nonneg__nonpos2,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),A2)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_4288_mult__nonneg__nonpos2,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),A2)),zero_zero_nat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_4289_mult__nonneg__nonpos2,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),A2)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_4290_zero__le__mult__iff,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),zero_zero_real)) ) ) ) ).

% zero_le_mult_iff
tff(fact_4291_zero__le__mult__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),zero_zero_rat)) ) ) ) ).

% zero_le_mult_iff
tff(fact_4292_zero__le__mult__iff,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),zero_zero_int)) ) ) ) ).

% zero_le_mult_iff
tff(fact_4293_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_4294_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_4295_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_4296_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_4297_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_4298_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_4299_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_4300_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_4301_mult__less__cancel__right__disj,axiom,
    ! [A2: real,C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_4302_mult__less__cancel__right__disj,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_4303_mult__less__cancel__right__disj,axiom,
    ! [A2: int,C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_4304_mult__strict__right__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C))) ) ) ).

% mult_strict_right_mono
tff(fact_4305_mult__strict__right__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C))) ) ) ).

% mult_strict_right_mono
tff(fact_4306_mult__strict__right__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),C))) ) ) ).

% mult_strict_right_mono
tff(fact_4307_mult__strict__right__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C))) ) ) ).

% mult_strict_right_mono
tff(fact_4308_mult__strict__right__mono__neg,axiom,
    ! [B3: real,A2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_4309_mult__strict__right__mono__neg,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_4310_mult__strict__right__mono__neg,axiom,
    ! [B3: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_4311_mult__less__cancel__left__disj,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_4312_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_4313_mult__less__cancel__left__disj,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_4314_mult__strict__left__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3))) ) ) ).

% mult_strict_left_mono
tff(fact_4315_mult__strict__left__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3))) ) ) ).

% mult_strict_left_mono
tff(fact_4316_mult__strict__left__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3))) ) ) ).

% mult_strict_left_mono
tff(fact_4317_mult__strict__left__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3))) ) ) ).

% mult_strict_left_mono
tff(fact_4318_mult__strict__left__mono__neg,axiom,
    ! [B3: real,A2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_4319_mult__strict__left__mono__neg,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_4320_mult__strict__left__mono__neg,axiom,
    ! [B3: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_4321_mult__less__cancel__left__pos,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_4322_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_4323_mult__less__cancel__left__pos,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_4324_mult__less__cancel__left__neg,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_4325_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_4326_mult__less__cancel__left__neg,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_4327_zero__less__mult__pos2,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),A2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3)) ) ) ).

% zero_less_mult_pos2
tff(fact_4328_zero__less__mult__pos2,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),A2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3)) ) ) ).

% zero_less_mult_pos2
tff(fact_4329_zero__less__mult__pos2,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),A2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3)) ) ) ).

% zero_less_mult_pos2
tff(fact_4330_zero__less__mult__pos2,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B3),A2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3)) ) ) ).

% zero_less_mult_pos2
tff(fact_4331_zero__less__mult__pos,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3)) ) ) ).

% zero_less_mult_pos
tff(fact_4332_zero__less__mult__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3)) ) ) ).

% zero_less_mult_pos
tff(fact_4333_zero__less__mult__pos,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3)) ) ) ).

% zero_less_mult_pos
tff(fact_4334_zero__less__mult__pos,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3)) ) ) ).

% zero_less_mult_pos
tff(fact_4335_zero__less__mult__iff,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),zero_zero_real)) ) ) ) ).

% zero_less_mult_iff
tff(fact_4336_zero__less__mult__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),zero_zero_rat)) ) ) ) ).

% zero_less_mult_iff
tff(fact_4337_zero__less__mult__iff,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),zero_zero_int)) ) ) ) ).

% zero_less_mult_iff
tff(fact_4338_mult__pos__neg2,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),A2)),zero_zero_real)) ) ) ).

% mult_pos_neg2
tff(fact_4339_mult__pos__neg2,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),A2)),zero_zero_rat)) ) ) ).

% mult_pos_neg2
tff(fact_4340_mult__pos__neg2,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),A2)),zero_zero_nat)) ) ) ).

% mult_pos_neg2
tff(fact_4341_mult__pos__neg2,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),A2)),zero_zero_int)) ) ) ).

% mult_pos_neg2
tff(fact_4342_mult__pos__pos,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3))) ) ) ).

% mult_pos_pos
tff(fact_4343_mult__pos__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3))) ) ) ).

% mult_pos_pos
tff(fact_4344_mult__pos__pos,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B3))) ) ) ).

% mult_pos_pos
tff(fact_4345_mult__pos__pos,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3))) ) ) ).

% mult_pos_pos
tff(fact_4346_mult__pos__neg,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)),zero_zero_real)) ) ) ).

% mult_pos_neg
tff(fact_4347_mult__pos__neg,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)),zero_zero_rat)) ) ) ).

% mult_pos_neg
tff(fact_4348_mult__pos__neg,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B3)),zero_zero_nat)) ) ) ).

% mult_pos_neg
tff(fact_4349_mult__pos__neg,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3)),zero_zero_int)) ) ) ).

% mult_pos_neg
tff(fact_4350_mult__neg__pos,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)),zero_zero_real)) ) ) ).

% mult_neg_pos
tff(fact_4351_mult__neg__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)),zero_zero_rat)) ) ) ).

% mult_neg_pos
tff(fact_4352_mult__neg__pos,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B3)),zero_zero_nat)) ) ) ).

% mult_neg_pos
tff(fact_4353_mult__neg__pos,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3)),zero_zero_int)) ) ) ).

% mult_neg_pos
tff(fact_4354_mult__less__0__iff,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3)) ) ) ) ).

% mult_less_0_iff
tff(fact_4355_mult__less__0__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3)) ) ) ) ).

% mult_less_0_iff
tff(fact_4356_mult__less__0__iff,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3)) ) ) ) ).

% mult_less_0_iff
tff(fact_4357_not__square__less__zero,axiom,
    ! [A2: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),A2)),zero_zero_real)) ).

% not_square_less_zero
tff(fact_4358_not__square__less__zero,axiom,
    ! [A2: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),A2)),zero_zero_rat)) ).

% not_square_less_zero
tff(fact_4359_not__square__less__zero,axiom,
    ! [A2: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),A2)),zero_zero_int)) ).

% not_square_less_zero
tff(fact_4360_mult__neg__neg,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3))) ) ) ).

% mult_neg_neg
tff(fact_4361_mult__neg__neg,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3))) ) ) ).

% mult_neg_neg
tff(fact_4362_mult__neg__neg,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3))) ) ) ).

% mult_neg_neg
tff(fact_4363_add__scale__eq__noteq,axiom,
    ! [R2: complex,A2: complex,B3: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A2 = B3 )
          & ( C != D ) )
       => ( aa_complex_complex(plus_plus_complex(A2),aa_complex_complex(times_times_complex(R2),C)) != aa_complex_complex(plus_plus_complex(B3),aa_complex_complex(times_times_complex(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4364_add__scale__eq__noteq,axiom,
    ! [R2: real,A2: real,B3: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A2 = B3 )
          & ( C != D ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),C)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4365_add__scale__eq__noteq,axiom,
    ! [R2: rat,A2: rat,B3: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A2 = B3 )
          & ( C != D ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),C)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4366_add__scale__eq__noteq,axiom,
    ! [R2: nat,A2: nat,B3: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A2 = B3 )
          & ( C != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4367_add__scale__eq__noteq,axiom,
    ! [R2: int,A2: int,B3: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A2 = B3 )
          & ( C != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4368_less__1__mult,axiom,
    ! [M: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),N))) ) ) ).

% less_1_mult
tff(fact_4369_less__1__mult,axiom,
    ! [M: rat,N: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),M))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M),N))) ) ) ).

% less_1_mult
tff(fact_4370_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% less_1_mult
tff(fact_4371_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M),N))) ) ) ).

% less_1_mult
tff(fact_4372_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z) )
        <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_4373_frac__eq__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_4374_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_complex_complex(divide1717551699836669952omplex(W2),Z) )
        <=> ( aa_complex_complex(times_times_complex(X),Z) = aa_complex_complex(times_times_complex(W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_4375_divide__eq__eq,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C) = A2 )
    <=> ( ( ( C != zero_zero_rat )
         => ( B3 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C) ) )
        & ( ( C = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
tff(fact_4376_divide__eq__eq,axiom,
    ! [B3: real,C: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C) = A2 )
    <=> ( ( ( C != zero_zero_real )
         => ( B3 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C) ) )
        & ( ( C = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% divide_eq_eq
tff(fact_4377_divide__eq__eq,axiom,
    ! [B3: complex,C: complex,A2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B3),C) = A2 )
    <=> ( ( ( C != zero_zero_complex )
         => ( B3 = aa_complex_complex(times_times_complex(A2),C) ) )
        & ( ( C = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
tff(fact_4378_eq__divide__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C) = B3 ) )
        & ( ( C = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
tff(fact_4379_eq__divide__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C) = B3 ) )
        & ( ( C = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% eq_divide_eq
tff(fact_4380_eq__divide__eq,axiom,
    ! [A2: complex,B3: complex,C: complex] :
      ( ( A2 = aa_complex_complex(divide1717551699836669952omplex(B3),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A2),C) = B3 ) )
        & ( ( C = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
tff(fact_4381_divide__eq__imp,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B3 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C) = A2 ) ) ) ).

% divide_eq_imp
tff(fact_4382_divide__eq__imp,axiom,
    ! [C: real,B3: real,A2: real] :
      ( ( C != zero_zero_real )
     => ( ( B3 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C) = A2 ) ) ) ).

% divide_eq_imp
tff(fact_4383_divide__eq__imp,axiom,
    ! [C: complex,B3: complex,A2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B3 = aa_complex_complex(times_times_complex(A2),C) )
       => ( aa_complex_complex(divide1717551699836669952omplex(B3),C) = A2 ) ) ) ).

% divide_eq_imp
tff(fact_4384_eq__divide__imp,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C) = B3 )
       => ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C) ) ) ) ).

% eq_divide_imp
tff(fact_4385_eq__divide__imp,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C) = B3 )
       => ( A2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C) ) ) ) ).

% eq_divide_imp
tff(fact_4386_eq__divide__imp,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A2),C) = B3 )
       => ( A2 = aa_complex_complex(divide1717551699836669952omplex(B3),C) ) ) ) ).

% eq_divide_imp
tff(fact_4387_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C) = A2 )
      <=> ( B3 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_4388_nonzero__divide__eq__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C) = A2 )
      <=> ( B3 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_4389_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B3: complex,A2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(B3),C) = A2 )
      <=> ( B3 = aa_complex_complex(times_times_complex(A2),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_4390_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C) = B3 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_4391_nonzero__eq__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C) = B3 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_4392_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A2 = aa_complex_complex(divide1717551699836669952omplex(B3),C) )
      <=> ( aa_complex_complex(times_times_complex(A2),C) = B3 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_4393_abs__mult__less,axiom,
    ! [A2: code_integer,C: code_integer,B3: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A2)),C))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(B3)),D))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,times_3573771949741848930nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B3))),times_3573771949741848930nteger(C,D))) ) ) ).

% abs_mult_less
tff(fact_4394_abs__mult__less,axiom,
    ! [A2: real,C: real,B3: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A2)),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,B3)),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D))) ) ) ).

% abs_mult_less
tff(fact_4395_abs__mult__less,axiom,
    ! [A2: rat,C: rat,B3: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A2)),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,B3)),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B3))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),D))) ) ) ).

% abs_mult_less
tff(fact_4396_abs__mult__less,axiom,
    ! [A2: int,C: int,B3: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A2)),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,B3)),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B3))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))) ) ) ).

% abs_mult_less
tff(fact_4397_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J))) ) ) ).

% zmult_zless_mono2
tff(fact_4398_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),U))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X))) ).

% real_minus_mult_self_le
tff(fact_4399_Gcd__int__greater__eq__0,axiom,
    ! [K4: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),gcd_Gcd_int(K4))) ).

% Gcd_int_greater_eq_0
tff(fact_4400_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).

% real_root_less_mono
tff(fact_4401_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).

% real_root_le_mono
tff(fact_4402_real__root__power,axiom,
    ! [N: nat,X: real,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),K) ) ) ).

% real_root_power
tff(fact_4403_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(abs_abs_real,X)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) ) ) ).

% real_root_abs
tff(fact_4404_log__base__root,axiom,
    ! [N: nat,B3: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3))
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,N),B3)),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B3),X)) ) ) ) ).

% log_base_root
tff(fact_4405_mult__less__le__imp__less,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_4406_mult__less__le__imp__less,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_4407_mult__less__le__imp__less,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_4408_mult__less__le__imp__less,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_4409_mult__le__less__imp__less,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_4410_mult__le__less__imp__less,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_4411_mult__le__less__imp__less,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_4412_mult__le__less__imp__less,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_4413_mult__right__le__imp__le,axiom,
    ! [A2: real,C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ) ).

% mult_right_le_imp_le
tff(fact_4414_mult__right__le__imp__le,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ) ).

% mult_right_le_imp_le
tff(fact_4415_mult__right__le__imp__le,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),C)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3)) ) ) ).

% mult_right_le_imp_le
tff(fact_4416_mult__right__le__imp__le,axiom,
    ! [A2: int,C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ) ).

% mult_right_le_imp_le
tff(fact_4417_mult__left__le__imp__le,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ) ).

% mult_left_le_imp_le
tff(fact_4418_mult__left__le__imp__le,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ) ).

% mult_left_le_imp_le
tff(fact_4419_mult__left__le__imp__le,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3)) ) ) ).

% mult_left_le_imp_le
tff(fact_4420_mult__left__le__imp__le,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ) ).

% mult_left_le_imp_le
tff(fact_4421_mult__le__cancel__left__pos,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_4422_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_4423_mult__le__cancel__left__pos,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_4424_mult__le__cancel__left__neg,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_4425_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_4426_mult__le__cancel__left__neg,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_4427_mult__less__cancel__right,axiom,
    ! [A2: real,C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2)) ) ) ) ).

% mult_less_cancel_right
tff(fact_4428_mult__less__cancel__right,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2)) ) ) ) ).

% mult_less_cancel_right
tff(fact_4429_mult__less__cancel__right,axiom,
    ! [A2: int,C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2)) ) ) ) ).

% mult_less_cancel_right
tff(fact_4430_mult__strict__mono_H,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_4431_mult__strict__mono_H,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_4432_mult__strict__mono_H,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_4433_mult__strict__mono_H,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_4434_mult__right__less__imp__less,axiom,
    ! [A2: real,C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) ) ) ).

% mult_right_less_imp_less
tff(fact_4435_mult__right__less__imp__less,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) ) ) ).

% mult_right_less_imp_less
tff(fact_4436_mult__right__less__imp__less,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),C)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3)) ) ) ).

% mult_right_less_imp_less
tff(fact_4437_mult__right__less__imp__less,axiom,
    ! [A2: int,C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3)) ) ) ).

% mult_right_less_imp_less
tff(fact_4438_mult__less__cancel__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2)) ) ) ) ).

% mult_less_cancel_left
tff(fact_4439_mult__less__cancel__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2)) ) ) ) ).

% mult_less_cancel_left
tff(fact_4440_mult__less__cancel__left,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2)) ) ) ) ).

% mult_less_cancel_left
tff(fact_4441_mult__strict__mono,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_4442_mult__strict__mono,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_4443_mult__strict__mono,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_4444_mult__strict__mono,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_4445_mult__left__less__imp__less,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) ) ) ).

% mult_left_less_imp_less
tff(fact_4446_mult__left__less__imp__less,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) ) ) ).

% mult_left_less_imp_less
tff(fact_4447_mult__left__less__imp__less,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3)) ) ) ).

% mult_left_less_imp_less
tff(fact_4448_mult__left__less__imp__less,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3)) ) ) ).

% mult_left_less_imp_less
tff(fact_4449_mult__le__cancel__right,axiom,
    ! [A2: real,C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2)) ) ) ) ).

% mult_le_cancel_right
tff(fact_4450_mult__le__cancel__right,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2)) ) ) ) ).

% mult_le_cancel_right
tff(fact_4451_mult__le__cancel__right,axiom,
    ! [A2: int,C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2)) ) ) ) ).

% mult_le_cancel_right
tff(fact_4452_mult__le__cancel__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2)) ) ) ) ).

% mult_le_cancel_left
tff(fact_4453_mult__le__cancel__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2)) ) ) ) ).

% mult_le_cancel_left
tff(fact_4454_mult__le__cancel__left,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2)) ) ) ) ).

% mult_le_cancel_left
tff(fact_4455_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real))
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_4456_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat))
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_4457_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int))
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_4458_sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_4459_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_4460_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_4461_mult__left__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_4462_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_4463_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_4464_mult__right__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_4465_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_4466_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_4467_mult__le__one,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)),one_one_real)) ) ) ) ).

% mult_le_one
tff(fact_4468_mult__le__one,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),one_one_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)),one_one_rat)) ) ) ) ).

% mult_le_one
tff(fact_4469_mult__le__one,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B3)),one_one_nat)) ) ) ) ).

% mult_le_one
tff(fact_4470_mult__le__one,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3)),one_one_int)) ) ) ) ).

% mult_le_one
tff(fact_4471_mult__left__le,axiom,
    ! [C: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),A2)) ) ) ).

% mult_left_le
tff(fact_4472_mult__left__le,axiom,
    ! [C: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),one_one_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),A2)) ) ) ).

% mult_left_le
tff(fact_4473_mult__left__le,axiom,
    ! [C: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),A2)) ) ) ).

% mult_left_le
tff(fact_4474_mult__left__le,axiom,
    ! [C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),A2)) ) ) ).

% mult_left_le
tff(fact_4475_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_4476_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_4477_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_4478_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)) ).

% not_sum_squares_lt_zero
tff(fact_4479_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat)) ).

% not_sum_squares_lt_zero
tff(fact_4480_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)) ).

% not_sum_squares_lt_zero
tff(fact_4481_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B3)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_4482_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_4483_divide__strict__left__mono__neg,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B3))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_4484_divide__strict__left__mono__neg,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B3))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_4485_divide__strict__left__mono,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B3))) ) ) ) ).

% divide_strict_left_mono
tff(fact_4486_divide__strict__left__mono,axiom,
    ! [B3: real,A2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B3))) ) ) ) ).

% divide_strict_left_mono
tff(fact_4487_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_4488_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_4489_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),Z)) ) ) ).

% mult_imp_div_pos_less
tff(fact_4490_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z)) ) ) ).

% mult_imp_div_pos_less
tff(fact_4491_pos__less__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),B3)) ) ) ).

% pos_less_divide_eq
tff(fact_4492_pos__less__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),B3)) ) ) ).

% pos_less_divide_eq
tff(fact_4493_pos__divide__less__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)),A2))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C))) ) ) ).

% pos_divide_less_eq
tff(fact_4494_pos__divide__less__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)),A2))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C))) ) ) ).

% pos_divide_less_eq
tff(fact_4495_neg__less__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C))) ) ) ).

% neg_less_divide_eq
tff(fact_4496_neg__less__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C))) ) ) ).

% neg_less_divide_eq
tff(fact_4497_neg__divide__less__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)),A2))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),B3)) ) ) ).

% neg_divide_less_eq
tff(fact_4498_neg__divide__less__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)),A2))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),B3)) ) ) ).

% neg_divide_less_eq
tff(fact_4499_less__divide__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),B3)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_4500_less__divide__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),B3)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_4501_divide__less__eq,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)),A2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),B3)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_4502_divide__less__eq,axiom,
    ! [B3: real,C: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)),A2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),B3)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_4503_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: real,E2: real,C: real,B3: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)),E2)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4504_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: rat,E2: rat,C: rat,B3: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)),E2)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4505_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: int,E2: int,C: int,B3: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)),E2)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4506_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: real,E2: real,C: real,B3: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),A2)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4507_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: rat,E2: rat,C: rat,B3: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B3),A2)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4508_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: int,E2: int,C: int,B3: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B3),A2)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4509_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A2: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),Z)),B3) = B3 ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),Z)),B3) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_4510_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: real,A2: real,B3: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),Z)),B3) = B3 ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),Z)),B3) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_4511_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: complex,A2: complex,B3: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A2),Z)),B3) = B3 ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A2),Z)),B3) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A2),aa_complex_complex(times_times_complex(B3),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_4512_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A2: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),Z)) = A2 ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),Z)),B3)),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_4513_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: real,A2: real,B3: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),Z)) = A2 ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),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,A2),Z)),B3)),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_4514_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: complex,A2: complex,B3: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(A2),aa_complex_complex(divide1717551699836669952omplex(B3),Z)) = A2 ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(A2),aa_complex_complex(divide1717551699836669952omplex(B3),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A2),Z)),B3)),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_4515_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_4516_add__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: 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,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),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,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_4517_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W2),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(W2),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_4518_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),Z) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_4519_add__frac__num,axiom,
    ! [Y: real,X: 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,X),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_4520_add__frac__num,axiom,
    ! [Y: complex,X: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_4521_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_4522_add__num__frac,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_4523_add__num__frac,axiom,
    ! [Y: complex,Z: complex,X: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_4524_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_4525_add__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),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,X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_4526_add__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_4527_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_4528_divide__add__eq__iff,axiom,
    ! [Z: real,X: 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,X),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_4529_divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_4530_less__add__iff1,axiom,
    ! [A2: real,E2: real,C: real,B3: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)),E2)),C)),D)) ) ).

% less_add_iff1
tff(fact_4531_less__add__iff1,axiom,
    ! [A2: rat,E2: rat,C: rat,B3: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B3)),E2)),C)),D)) ) ).

% less_add_iff1
tff(fact_4532_less__add__iff1,axiom,
    ! [A2: int,E2: int,C: int,B3: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B3)),E2)),C)),D)) ) ).

% less_add_iff1
tff(fact_4533_less__add__iff2,axiom,
    ! [A2: real,E2: real,C: real,B3: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),A2)),E2)),D))) ) ).

% less_add_iff2
tff(fact_4534_less__add__iff2,axiom,
    ! [A2: rat,E2: rat,C: rat,B3: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B3),A2)),E2)),D))) ) ).

% less_add_iff2
tff(fact_4535_less__add__iff2,axiom,
    ! [A2: int,E2: int,C: int,B3: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B3),A2)),E2)),D))) ) ).

% less_add_iff2
tff(fact_4536_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A2: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),Z)) = A2 ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),Z)),B3)),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_4537_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: real,A2: real,B3: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),Z)) = A2 ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),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,A2),Z)),B3)),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_4538_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: complex,A2: complex,B3: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(A2),aa_complex_complex(divide1717551699836669952omplex(B3),Z)) = A2 ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(A2),aa_complex_complex(divide1717551699836669952omplex(B3),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A2),Z)),B3)),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_4539_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4540_diff__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: 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,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),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,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4541_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W2),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(W2),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4542_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4543_diff__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),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,X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4544_diff__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4545_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4546_divide__diff__eq__iff,axiom,
    ! [Z: real,X: 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,X),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4547_divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(X),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4548_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),X))) ) ).

% ex_less_of_nat_mult
tff(fact_4549_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),X))) ) ).

% ex_less_of_nat_mult
tff(fact_4550_eq__minus__divide__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( A2 = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C) = aa_real_real(uminus_uminus_real,B3) ) )
        & ( ( C = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_4551_eq__minus__divide__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( A2 = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C) = aa_rat_rat(uminus_uminus_rat,B3) ) )
        & ( ( C = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_4552_eq__minus__divide__eq,axiom,
    ! [A2: complex,B3: complex,C: complex] :
      ( ( A2 = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B3),C)) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A2),C) = aa_complex_complex(uminus1482373934393186551omplex,B3) ) )
        & ( ( C = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_4553_minus__divide__eq__eq,axiom,
    ! [B3: real,C: real,A2: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)) = A2 )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,B3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C) ) )
        & ( ( C = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_4554_minus__divide__eq__eq,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)) = A2 )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,B3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C) ) )
        & ( ( C = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_4555_minus__divide__eq__eq,axiom,
    ! [B3: complex,C: complex,A2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B3),C)) = A2 )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,B3) = aa_complex_complex(times_times_complex(A2),C) ) )
        & ( ( C = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_4556_nonzero__neg__divide__eq__eq,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( B3 != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3)) = C )
      <=> ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_4557_nonzero__neg__divide__eq__eq,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3)) = C )
      <=> ( aa_rat_rat(uminus_uminus_rat,A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_4558_nonzero__neg__divide__eq__eq,axiom,
    ! [B3: complex,A2: complex,C: complex] :
      ( ( B3 != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A2),B3)) = C )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A2) = aa_complex_complex(times_times_complex(C),B3) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_4559_nonzero__neg__divide__eq__eq2,axiom,
    ! [B3: real,C: real,A2: real] :
      ( ( B3 != zero_zero_real )
     => ( ( C = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3)) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3) = aa_real_real(uminus_uminus_real,A2) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_4560_nonzero__neg__divide__eq__eq2,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( C = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),B3)) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3) = aa_rat_rat(uminus_uminus_rat,A2) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_4561_nonzero__neg__divide__eq__eq2,axiom,
    ! [B3: complex,C: complex,A2: complex] :
      ( ( B3 != zero_zero_complex )
     => ( ( C = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A2),B3)) )
      <=> ( aa_complex_complex(times_times_complex(C),B3) = aa_complex_complex(uminus1482373934393186551omplex,A2) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_4562_power__gt1__lemma,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)))) ) ).

% power_gt1_lemma
tff(fact_4563_power__gt1__lemma,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_nat_rat(power_power_rat(A2),N)))) ) ).

% power_gt1_lemma
tff(fact_4564_power__gt1__lemma,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)))) ) ).

% power_gt1_lemma
tff(fact_4565_power__gt1__lemma,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(power_power_int(A2),N)))) ) ).

% power_gt1_lemma
tff(fact_4566_power__less__power__Suc,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)))) ) ).

% power_less_power_Suc
tff(fact_4567_power__less__power__Suc,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_nat_rat(power_power_rat(A2),N)))) ) ).

% power_less_power_Suc
tff(fact_4568_power__less__power__Suc,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)))) ) ).

% power_less_power_Suc
tff(fact_4569_power__less__power__Suc,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A2),N)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(power_power_int(A2),N)))) ) ).

% power_less_power_Suc
tff(fact_4570_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
     => ( times_3573771949741848930nteger(abs_abs_Code_integer(Y),X) = abs_abs_Code_integer(times_3573771949741848930nteger(Y,X)) ) ) ).

% abs_mult_pos
tff(fact_4571_abs__mult__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Y)),X) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_4572_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,Y)),X) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_4573_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),X) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_4574_abs__eq__mult,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
          | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B3))
          | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B3),zero_z3403309356797280102nteger)) ) )
     => ( abs_abs_Code_integer(times_3573771949741848930nteger(A2,B3)) = times_3573771949741848930nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B3)) ) ) ).

% abs_eq_mult
tff(fact_4575_abs__eq__mult,axiom,
    ! [A2: real,B3: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
          | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3))
          | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),zero_zero_real)) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B3)) ) ) ).

% abs_eq_mult
tff(fact_4576_abs__eq__mult,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
          | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3))
          | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),zero_zero_rat)) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B3)) ) ) ).

% abs_eq_mult
tff(fact_4577_abs__eq__mult,axiom,
    ! [A2: int,B3: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3))
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),zero_zero_int)) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B3)) ) ) ).

% abs_eq_mult
tff(fact_4578_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_4579_reals__Archimedean3,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ! [Y4: real] :
        ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),X))) ) ).

% reals_Archimedean3
tff(fact_4580_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_4581_minusinfinity,axiom,
    ! [D: int,P1: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X5: int,K2: int] :
            ( pp(aa_int_bool(P1,X5))
          <=> pp(aa_int_bool(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) )
       => ( ? [Z4: int] :
            ! [X5: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z4))
             => ( pp(aa_int_bool(P,X5))
              <=> pp(aa_int_bool(P1,X5)) ) )
         => ( ? [X_13: int] : pp(aa_int_bool(P1,X_13))
           => ? [X_12: int] : pp(aa_int_bool(P,X_12)) ) ) ) ) ).

% minusinfinity
tff(fact_4582_plusinfinity,axiom,
    ! [D: int,P2: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X5: int,K2: int] :
            ( pp(aa_int_bool(P2,X5))
          <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) )
       => ( ? [Z4: int] :
            ! [X5: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X5))
             => ( pp(aa_int_bool(P,X5))
              <=> pp(aa_int_bool(P2,X5)) ) )
         => ( ? [X_13: int] : pp(aa_int_bool(P2,X_13))
           => ? [X_12: int] : pp(aa_int_bool(P,X_12)) ) ) ) ) ).

% plusinfinity
tff(fact_4583_zdiv__zmult2__eq,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_4584_find__None__iff2,axiom,
    ! [P: fun_real_bool,Xs: list_real] :
      ( ( none_real = find_real(P,Xs) )
    <=> ~ ? [X3: real] :
            ( pp(aa_set_real_bool(member_real(X3),set_real2(Xs)))
            & pp(aa_real_bool(P,X3)) ) ) ).

% find_None_iff2
tff(fact_4585_find__None__iff2,axiom,
    ! [P: fun_bool_bool,Xs: list_bool] :
      ( ( none_bool = find_bool(P,Xs) )
    <=> ~ ? [X3: bool] :
            ( pp(aa_set_bool_bool(member_bool(X3),set_bool2(Xs)))
            & pp(aa_bool_bool(P,X3)) ) ) ).

% find_None_iff2
tff(fact_4586_find__None__iff2,axiom,
    ! [P: fun_set_nat_bool,Xs: list_set_nat] :
      ( ( none_set_nat = find_set_nat(P,Xs) )
    <=> ~ ? [X3: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(Xs)))
            & pp(aa_set_nat_bool(P,X3)) ) ) ).

% find_None_iff2
tff(fact_4587_find__None__iff2,axiom,
    ! [P: fun_int_bool,Xs: list_int] :
      ( ( none_int = find_int(P,Xs) )
    <=> ~ ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
            & pp(aa_int_bool(P,X3)) ) ) ).

% find_None_iff2
tff(fact_4588_find__None__iff2,axiom,
    ! [P: fun_VEBT_VEBT_bool,Xs: list_VEBT_VEBT] :
      ( ( none_VEBT_VEBT = find_VEBT_VEBT(P,Xs) )
    <=> ~ ? [X3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
            & pp(aa_VEBT_VEBT_bool(P,X3)) ) ) ).

% find_None_iff2
tff(fact_4589_find__None__iff2,axiom,
    ! [P: fun_nat_bool,Xs: list_nat] :
      ( ( none_nat = find_nat(P,Xs) )
    <=> ~ ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
            & pp(aa_nat_bool(P,X3)) ) ) ).

% find_None_iff2
tff(fact_4590_find__None__iff2,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,Xs: list_P6011104703257516679at_nat] :
      ( ( none_P5556105721700978146at_nat = find_P8199882355184865565at_nat(P,Xs) )
    <=> ~ ? [X3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),set_Pr5648618587558075414at_nat(Xs)))
            & pp(aa_Pro2232559814647311250t_bool(P,X3)) ) ) ).

% find_None_iff2
tff(fact_4591_find__None__iff2,axiom,
    ! [P: fun_num_bool,Xs: list_num] :
      ( ( none_num = find_num(P,Xs) )
    <=> ~ ? [X3: num] :
            ( pp(aa_set_num_bool(member_num(X3),set_num2(Xs)))
            & pp(aa_num_bool(P,X3)) ) ) ).

% find_None_iff2
tff(fact_4592_find__None__iff,axiom,
    ! [P: fun_real_bool,Xs: list_real] :
      ( ( find_real(P,Xs) = none_real )
    <=> ~ ? [X3: real] :
            ( pp(aa_set_real_bool(member_real(X3),set_real2(Xs)))
            & pp(aa_real_bool(P,X3)) ) ) ).

% find_None_iff
tff(fact_4593_find__None__iff,axiom,
    ! [P: fun_bool_bool,Xs: list_bool] :
      ( ( find_bool(P,Xs) = none_bool )
    <=> ~ ? [X3: bool] :
            ( pp(aa_set_bool_bool(member_bool(X3),set_bool2(Xs)))
            & pp(aa_bool_bool(P,X3)) ) ) ).

% find_None_iff
tff(fact_4594_find__None__iff,axiom,
    ! [P: fun_set_nat_bool,Xs: list_set_nat] :
      ( ( find_set_nat(P,Xs) = none_set_nat )
    <=> ~ ? [X3: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(Xs)))
            & pp(aa_set_nat_bool(P,X3)) ) ) ).

% find_None_iff
tff(fact_4595_find__None__iff,axiom,
    ! [P: fun_int_bool,Xs: list_int] :
      ( ( find_int(P,Xs) = none_int )
    <=> ~ ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
            & pp(aa_int_bool(P,X3)) ) ) ).

% find_None_iff
tff(fact_4596_find__None__iff,axiom,
    ! [P: fun_VEBT_VEBT_bool,Xs: list_VEBT_VEBT] :
      ( ( find_VEBT_VEBT(P,Xs) = none_VEBT_VEBT )
    <=> ~ ? [X3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
            & pp(aa_VEBT_VEBT_bool(P,X3)) ) ) ).

% find_None_iff
tff(fact_4597_find__None__iff,axiom,
    ! [P: fun_nat_bool,Xs: list_nat] :
      ( ( find_nat(P,Xs) = none_nat )
    <=> ~ ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
            & pp(aa_nat_bool(P,X3)) ) ) ).

% find_None_iff
tff(fact_4598_find__None__iff,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,Xs: list_P6011104703257516679at_nat] :
      ( ( find_P8199882355184865565at_nat(P,Xs) = none_P5556105721700978146at_nat )
    <=> ~ ? [X3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),set_Pr5648618587558075414at_nat(Xs)))
            & pp(aa_Pro2232559814647311250t_bool(P,X3)) ) ) ).

% find_None_iff
tff(fact_4599_find__None__iff,axiom,
    ! [P: fun_num_bool,Xs: list_num] :
      ( ( find_num(P,Xs) = none_num )
    <=> ~ ? [X3: num] :
            ( pp(aa_set_num_bool(member_num(X3),set_num2(Xs)))
            & pp(aa_num_bool(P,X3)) ) ) ).

% find_None_iff
tff(fact_4600_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ).

% real_root_gt_zero
tff(fact_4601_real__root__strict__decreasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N4),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_4602_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_4603_mult__less__cancel__right2,axiom,
    ! [A2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_4604_mult__less__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_4605_mult__less__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_4606_mult__less__cancel__right1,axiom,
    ! [C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),one_one_real)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_4607_mult__less__cancel__right1,axiom,
    ! [C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B3)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),one_one_rat)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_4608_mult__less__cancel__right1,axiom,
    ! [C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B3)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),one_one_int)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_4609_mult__less__cancel__left2,axiom,
    ! [C: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_4610_mult__less__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_4611_mult__less__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_4612_mult__less__cancel__left1,axiom,
    ! [C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),one_one_real)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_4613_mult__less__cancel__left1,axiom,
    ! [C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B3)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),one_one_rat)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_4614_mult__less__cancel__left1,axiom,
    ! [C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B3)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),one_one_int)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_4615_mult__le__cancel__right2,axiom,
    ! [A2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A2)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_4616_mult__le__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A2)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_4617_mult__le__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A2)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_4618_mult__le__cancel__right1,axiom,
    ! [C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B3)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),one_one_real)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_4619_mult__le__cancel__right1,axiom,
    ! [C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B3)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),one_one_rat)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_4620_mult__le__cancel__right1,axiom,
    ! [C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B3)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),one_one_int)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_4621_mult__le__cancel__left2,axiom,
    ! [C: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A2)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A2)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_4622_mult__le__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A2)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_4623_mult__le__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A2)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_4624_mult__le__cancel__left1,axiom,
    ! [C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B3)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B3)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),one_one_real)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_4625_mult__le__cancel__left1,axiom,
    ! [C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B3)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B3)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),one_one_rat)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_4626_mult__le__cancel__left1,axiom,
    ! [C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B3)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B3)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),one_one_int)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_4627_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z3),X)),Y)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_4628_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z3))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),one_one_rat))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z3),X)),Y)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_4629_divide__left__mono__neg,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B3))) ) ) ) ).

% divide_left_mono_neg
tff(fact_4630_divide__left__mono__neg,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B3))) ) ) ) ).

% divide_left_mono_neg
tff(fact_4631_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_4632_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_4633_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z)) ) ) ).

% mult_imp_div_pos_le
tff(fact_4634_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),Z)) ) ) ).

% mult_imp_div_pos_le
tff(fact_4635_pos__le__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),B3)) ) ) ).

% pos_le_divide_eq
tff(fact_4636_pos__le__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),B3)) ) ) ).

% pos_le_divide_eq
tff(fact_4637_pos__divide__le__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)),A2))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C))) ) ) ).

% pos_divide_le_eq
tff(fact_4638_pos__divide__le__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)),A2))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C))) ) ) ).

% pos_divide_le_eq
tff(fact_4639_neg__le__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C))) ) ) ).

% neg_le_divide_eq
tff(fact_4640_neg__le__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C))) ) ) ).

% neg_le_divide_eq
tff(fact_4641_neg__divide__le__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)),A2))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),B3)) ) ) ).

% neg_divide_le_eq
tff(fact_4642_neg__divide__le__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)),A2))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),B3)) ) ) ).

% neg_divide_le_eq
tff(fact_4643_divide__left__mono,axiom,
    ! [B3: real,A2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B3))) ) ) ) ).

% divide_left_mono
tff(fact_4644_divide__left__mono,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B3))) ) ) ) ).

% divide_left_mono
tff(fact_4645_le__divide__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),B3)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq
tff(fact_4646_le__divide__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),B3)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq
tff(fact_4647_divide__le__eq,axiom,
    ! [B3: real,C: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)),A2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),B3)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2)) ) ) ) ) ) ).

% divide_le_eq
tff(fact_4648_divide__le__eq,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)),A2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),B3)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2)) ) ) ) ) ) ).

% divide_le_eq
tff(fact_4649_convex__bound__le,axiom,
    ! [X: real,A2: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),A2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A2)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_4650_convex__bound__le,axiom,
    ! [X: rat,A2: rat,Y: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),A2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A2)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_4651_convex__bound__le,axiom,
    ! [X: int,A2: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A2)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_4652_frac__le__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real)) ) ) ) ).

% frac_le_eq
tff(fact_4653_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat)) ) ) ) ).

% frac_le_eq
tff(fact_4654_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat)) ) ) ) ).

% frac_less_eq
tff(fact_4655_frac__less__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real)) ) ) ) ).

% frac_less_eq
tff(fact_4656_power__Suc__less,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ) ).

% power_Suc_less
tff(fact_4657_power__Suc__less,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_nat_rat(power_power_rat(A2),N))),aa_nat_rat(power_power_rat(A2),N))) ) ) ).

% power_Suc_less
tff(fact_4658_power__Suc__less,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ) ).

% power_Suc_less
tff(fact_4659_power__Suc__less,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(power_power_int(A2),N))),aa_nat_int(power_power_int(A2),N))) ) ) ).

% power_Suc_less
tff(fact_4660_pos__minus__divide__less__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C))),A2))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_4661_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C))),A2))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_4662_pos__less__minus__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(uminus_uminus_real,B3))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_4663_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(uminus_uminus_rat,B3))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_4664_neg__minus__divide__less__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C))),A2))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(uminus_uminus_real,B3))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_4665_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C))),A2))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(uminus_uminus_rat,B3))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_4666_neg__less__minus__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_4667_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_4668_minus__divide__less__eq,axiom,
    ! [B3: real,C: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C))),A2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(uminus_uminus_real,B3))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2)) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_4669_minus__divide__less__eq,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C))),A2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(uminus_uminus_rat,B3))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2)) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_4670_less__minus__divide__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(uminus_uminus_real,B3))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_4671_less__minus__divide__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(uminus_uminus_rat,B3))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_4672_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: real,A2: real,B3: 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,A2),Z))),B3) = B3 ) )
      & ( ( 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,A2),Z))),B3) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_4673_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A2: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),Z))),B3) = B3 ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),Z))),B3) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_4674_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: complex,A2: complex,B3: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A2),Z))),B3) = B3 ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A2),Z))),B3) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(times_times_complex(B3),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_4675_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X: 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,X),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,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4676_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z))),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4677_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4678_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: real,A2: real,B3: 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,A2),Z))),B3) = aa_real_real(uminus_uminus_real,B3) ) )
      & ( ( 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,A2),Z))),B3) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_4679_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A2: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),Z))),B3) = aa_rat_rat(uminus_uminus_rat,B3) ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),Z))),B3) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_4680_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: complex,A2: complex,B3: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A2),Z))),B3) = aa_complex_complex(uminus1482373934393186551omplex,B3) ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A2),Z))),B3) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(times_times_complex(B3),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_4681_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: real,A2: real,B3: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),Z)),B3) = aa_real_real(uminus_uminus_real,B3) ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),Z)),B3) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_4682_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A2: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),Z)),B3) = aa_rat_rat(uminus_uminus_rat,B3) ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),Z)),B3) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B3),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_4683_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: complex,A2: complex,B3: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A2),Z)),B3) = aa_complex_complex(uminus1482373934393186551omplex,B3) ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A2),Z)),B3) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A2),aa_complex_complex(times_times_complex(B3),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_4684_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X: 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,X),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,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4685_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z))),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4686_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4687_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_4688_ln__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(ln_ln_real,X)) ) ) ).

% ln_realpow
tff(fact_4689_q__pos__lemma,axiom,
    ! [B5: int,Q5: int,R4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R4)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B5))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Q5)) ) ) ) ).

% q_pos_lemma
tff(fact_4690_zdiv__mono2__lemma,axiom,
    ! [B3: int,Q3: int,R2: int,B5: int,Q5: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q3)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R4)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B5))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),B3))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q3),Q5)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_4691_zdiv__mono2__neg__lemma,axiom,
    ! [B3: int,Q3: int,R2: int,B5: int,Q5: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q3)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R4)),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B3))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R4))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),B3))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q5),Q3)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_4692_unique__quotient__lemma,axiom,
    ! [B3: int,Q5: int,R4: int,Q3: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q3)),R2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R4))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B3))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q5),Q3)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_4693_unique__quotient__lemma__neg,axiom,
    ! [B3: int,Q5: int,R4: int,Q3: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q3)),R2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),R2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),R4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q3),Q5)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_4694_incr__mult__lemma,axiom,
    ! [D: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X5: int] :
            ( pp(aa_int_bool(P,X5))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X4: int] :
              ( pp(aa_int_bool(P,X4))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_4695_decr__mult__lemma,axiom,
    ! [D: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X5: int] :
            ( pp(aa_int_bool(P,X5))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X4: int] :
              ( pp(aa_int_bool(P,X4))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_4696_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ).

% real_root_pos_pos
tff(fact_4697_real__root__strict__increasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N4),X))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_4698_real__root__decreasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N4),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).

% real_root_decreasing
tff(fact_4699_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ) ).

% real_root_pow_pos
tff(fact_4700_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N) = X )
         => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_4701_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_4702_convex__bound__lt,axiom,
    ! [X: real,A2: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),A2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A2)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_4703_convex__bound__lt,axiom,
    ! [X: rat,A2: rat,Y: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),A2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A2)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_4704_convex__bound__lt,axiom,
    ! [X: int,A2: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A2)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_4705_le__minus__divide__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(uminus_uminus_real,B3))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_4706_le__minus__divide__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(uminus_uminus_rat,B3))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_4707_minus__divide__le__eq,axiom,
    ! [B3: real,C: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C))),A2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(uminus_uminus_real,B3))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2)) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_4708_minus__divide__le__eq,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C))),A2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(uminus_uminus_rat,B3))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2)) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_4709_neg__le__minus__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_4710_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_4711_neg__minus__divide__le__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C))),A2))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(uminus_uminus_real,B3))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_4712_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C))),A2))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(uminus_uminus_rat,B3))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_4713_pos__le__minus__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C)),aa_real_real(uminus_uminus_real,B3))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_4714_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(uminus_uminus_rat,B3))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_4715_pos__minus__divide__le__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C))),A2))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_4716_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C))),A2))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_4717_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,U),V))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),R2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),S2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,V),U))),S2))),V)) ) ) ) ).

% scaling_mono
tff(fact_4718_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,U),V))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),R2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,R2),S2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,V),U))),S2))),V)) ) ) ) ).

% scaling_mono
tff(fact_4719_power__eq__if,axiom,
    ! [M: nat,P3: complex] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P3),M) = one_one_complex ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P3),M) = aa_complex_complex(times_times_complex(P3),aa_nat_complex(power_power_complex(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_4720_power__eq__if,axiom,
    ! [M: nat,P3: real] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,P3),M) = one_one_real ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,P3),M) = 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_4721_power__eq__if,axiom,
    ! [M: nat,P3: rat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P3),M) = one_one_rat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P3),M) = aa_rat_rat(aa_rat_fun_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_4722_power__eq__if,axiom,
    ! [M: nat,P3: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = one_one_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_4723_power__eq__if,axiom,
    ! [M: nat,P3: int] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_int(power_power_int(P3),M) = one_one_int ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_int(power_power_int(P3),M) = aa_int_int(aa_int_fun_int_int(times_times_int,P3),aa_nat_int(power_power_int(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_4724_power__minus__mult,axiom,
    ! [N: nat,A2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A2) = aa_nat_complex(power_power_complex(A2),N) ) ) ).

% power_minus_mult
tff(fact_4725_power__minus__mult,axiom,
    ! [N: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) ) ) ).

% power_minus_mult
tff(fact_4726_power__minus__mult,axiom,
    ! [N: nat,A2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A2) = aa_nat_rat(power_power_rat(A2),N) ) ) ).

% power_minus_mult
tff(fact_4727_power__minus__mult,axiom,
    ! [N: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) ) ) ).

% power_minus_mult
tff(fact_4728_power__minus__mult,axiom,
    ! [N: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A2) = aa_nat_int(power_power_int(A2),N) ) ) ).

% power_minus_mult
tff(fact_4729_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => ( ! [M4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M4)),X)),C)) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_4730_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B3: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3))
         => ( ( B3 != one_one_real )
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B3)),aa_real_real(ln_ln_real,A2))),aa_real_real(aa_rea7340605660300375871l_real(log,B3),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_4731_log__mult,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A2),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_4732_split__zdiv,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)))
    <=> ( ( ( K = zero_zero_int )
         => pp(aa_int_bool(P,zero_zero_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
         => ! [I4: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => pp(aa_int_bool(P,I4)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I4: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => pp(aa_int_bool(P,I4)) ) ) ) ) ).

% split_zdiv
tff(fact_4733_int__div__neg__eq,axiom,
    ! [A2: int,B3: int,Q3: int,R2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q3)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),R2))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3) = Q3 ) ) ) ) ).

% int_div_neg_eq
tff(fact_4734_int__div__pos__eq,axiom,
    ! [A2: int,B3: int,Q3: int,R2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q3)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B3))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3) = Q3 ) ) ) ) ).

% int_div_pos_eq
tff(fact_4735_log__nat__power,axiom,
    ! [X: real,B3: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B3),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B3),X)) ) ) ).

% log_nat_power
tff(fact_4736_real__root__increasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N4),X))) ) ) ) ) ).

% real_root_increasing
tff(fact_4737_ln__root,axiom,
    ! [N: nat,B3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,N),B3)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B3)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% ln_root
tff(fact_4738_mult__le__cancel__iff2,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_4739_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_4740_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_4741_mult__le__cancel__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_4742_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_4743_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_4744_mult__less__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_4745_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_4746_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_4747_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)))) ) ) ) ).

% arctan_add
tff(fact_4748_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% root_powr_inverse
tff(fact_4749_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),product_Pair_int_int(Q3,R2)))
    <=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q3)),R2) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),L)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),R2))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int)) ) )
            & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( Q3 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_4750_log__minus__eq__powr,axiom,
    ! [B3: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3))
     => ( ( B3 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B3),X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B3),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_4751_split__root,axiom,
    ! [P: fun_real_bool,N: nat,X: real] :
      ( pp(aa_real_bool(P,aa_real_real(aa_nat_fun_real_real(root,N),X)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_real_bool(P,zero_zero_real)) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ! [Y2: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y2)),N)) = X )
             => pp(aa_real_bool(P,Y2)) ) ) ) ) ).

% split_root
tff(fact_4752_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_4753_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_4754_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: 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),N) )
    <=> ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_4755_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: 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,N),K) )
    <=> ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_4756_sgn__0,axiom,
    sgn_sgn_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% sgn_0
tff(fact_4757_sgn__0,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_0
tff(fact_4758_sgn__0,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_0
tff(fact_4759_sgn__0,axiom,
    aa_rat_rat(sgn_sgn_rat,zero_zero_rat) = zero_zero_rat ).

% sgn_0
tff(fact_4760_sgn__0,axiom,
    aa_int_int(sgn_sgn_int,zero_zero_int) = zero_zero_int ).

% sgn_0
tff(fact_4761_powr__0,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,zero_zero_real),Z) = zero_zero_real ).

% powr_0
tff(fact_4762_powr__eq__0__iff,axiom,
    ! [W2: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,W2),Z) = zero_zero_real )
    <=> ( W2 = zero_zero_real ) ) ).

% powr_eq_0_iff
tff(fact_4763_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_4764_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_4765_sgn__less,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,sgn_sgn_Code_integer(A2)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger)) ) ).

% sgn_less
tff(fact_4766_sgn__less,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sgn_sgn_real,A2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ).

% sgn_less
tff(fact_4767_sgn__less,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(sgn_sgn_rat,A2)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ).

% sgn_less
tff(fact_4768_sgn__less,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(sgn_sgn_int,A2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ).

% sgn_less
tff(fact_4769_sgn__greater,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),sgn_sgn_Code_integer(A2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)) ) ).

% sgn_greater
tff(fact_4770_sgn__greater,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sgn_sgn_real,A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2)) ) ).

% sgn_greater
tff(fact_4771_sgn__greater,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(sgn_sgn_rat,A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2)) ) ).

% sgn_greater
tff(fact_4772_sgn__greater,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(sgn_sgn_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2)) ) ).

% sgn_greater
tff(fact_4773_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_4774_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_4775_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% mult_less_cancel2
tff(fact_4776_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_4777_powr__zero__eq__one,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),zero_zero_real) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),zero_zero_real) = one_one_real ) ) ) ).

% powr_zero_eq_one
tff(fact_4778_mult__Suc__right,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc_right
tff(fact_4779_powr__gt__zero,axiom,
    ! [X: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2)))
    <=> ( X != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_4780_powr__nonneg__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),X)),zero_zero_real))
    <=> ( A2 = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_4781_powr__less__cancel__iff,axiom,
    ! [X: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B3)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) ) ) ).

% powr_less_cancel_iff
tff(fact_4782_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arctan,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% zero_less_arctan_iff
tff(fact_4783_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% arctan_less_zero_iff
tff(fact_4784_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% zero_le_arctan_iff
tff(fact_4785_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% arctan_le_zero_iff
tff(fact_4786_sgn__pos,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
     => ( sgn_sgn_Code_integer(A2) = one_one_Code_integer ) ) ).

% sgn_pos
tff(fact_4787_sgn__pos,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( aa_real_real(sgn_sgn_real,A2) = one_one_real ) ) ).

% sgn_pos
tff(fact_4788_sgn__pos,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( aa_rat_rat(sgn_sgn_rat,A2) = one_one_rat ) ) ).

% sgn_pos
tff(fact_4789_sgn__pos,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( aa_int_int(sgn_sgn_int,A2) = one_one_int ) ) ).

% sgn_pos
tff(fact_4790_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ) ) ).

% one_le_mult_iff
tff(fact_4791_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% mult_le_cancel2
tff(fact_4792_abs__sgn__eq__1,axiom,
    ! [A2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( abs_abs_Code_integer(sgn_sgn_Code_integer(A2)) = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
tff(fact_4793_abs__sgn__eq__1,axiom,
    ! [A2: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A2)) = one_one_real ) ) ).

% abs_sgn_eq_1
tff(fact_4794_abs__sgn__eq__1,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A2)) = one_one_rat ) ) ).

% abs_sgn_eq_1
tff(fact_4795_abs__sgn__eq__1,axiom,
    ! [A2: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A2)) = one_one_int ) ) ).

% abs_sgn_eq_1
tff(fact_4796_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),N) = M ) ) ).

% div_mult_self_is_m
tff(fact_4797_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),N) = M ) ) ).

% div_mult_self1_is_m
tff(fact_4798_powr__eq__one__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),X) = one_one_real )
      <=> ( X = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_4799_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),one_one_real) = X )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% powr_one_gt_zero_iff
tff(fact_4800_powr__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),one_one_real) = X ) ) ).

% powr_one
tff(fact_4801_powr__le__cancel__iff,axiom,
    ! [X: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B3)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ) ).

% powr_le_cancel_iff
tff(fact_4802_sgn__neg,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( aa_int_int(sgn_sgn_int,A2) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% sgn_neg
tff(fact_4803_sgn__neg,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( aa_real_real(sgn_sgn_real,A2) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% sgn_neg
tff(fact_4804_sgn__neg,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( aa_rat_rat(sgn_sgn_rat,A2) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% sgn_neg
tff(fact_4805_sgn__neg,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger))
     => ( sgn_sgn_Code_integer(A2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% sgn_neg
tff(fact_4806_powr__log__cancel,axiom,
    ! [A2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),aa_real_real(aa_rea7340605660300375871l_real(log,A2),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_4807_log__powr__cancel,axiom,
    ! [A2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_4808_sgn__0__0,axiom,
    ! [A2: code_integer] :
      ( ( sgn_sgn_Code_integer(A2) = zero_z3403309356797280102nteger )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% sgn_0_0
tff(fact_4809_sgn__0__0,axiom,
    ! [A2: real] :
      ( ( aa_real_real(sgn_sgn_real,A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% sgn_0_0
tff(fact_4810_sgn__0__0,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% sgn_0_0
tff(fact_4811_sgn__0__0,axiom,
    ! [A2: int] :
      ( ( aa_int_int(sgn_sgn_int,A2) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% sgn_0_0
tff(fact_4812_sgn__eq__0__iff,axiom,
    ! [A2: code_integer] :
      ( ( sgn_sgn_Code_integer(A2) = zero_z3403309356797280102nteger )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% sgn_eq_0_iff
tff(fact_4813_sgn__eq__0__iff,axiom,
    ! [A2: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,A2) = zero_zero_complex )
    <=> ( A2 = zero_zero_complex ) ) ).

% sgn_eq_0_iff
tff(fact_4814_sgn__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( aa_real_real(sgn_sgn_real,A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% sgn_eq_0_iff
tff(fact_4815_sgn__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% sgn_eq_0_iff
tff(fact_4816_sgn__eq__0__iff,axiom,
    ! [A2: int] :
      ( ( aa_int_int(sgn_sgn_int,A2) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% sgn_eq_0_iff
tff(fact_4817_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: 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)),N) )
    <=> ( M = N ) ) ).

% Suc_mult_cancel1
tff(fact_4818_mult__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N) = zero_zero_nat ).

% mult_0
tff(fact_4819_le__cube,axiom,
    ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)))) ).

% le_cube
tff(fact_4820_le__square,axiom,
    ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).

% le_square
tff(fact_4821_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L))) ) ) ).

% mult_le_mono
tff(fact_4822_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K))) ) ).

% mult_le_mono1
tff(fact_4823_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J))) ) ).

% mult_le_mono2
tff(fact_4824_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y))) ) ).

% arctan_monotone
tff(fact_4825_arctan__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% arctan_less_iff
tff(fact_4826_arctan__monotone_H,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y))) ) ).

% arctan_monotone'
tff(fact_4827_arctan__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% arctan_le_iff
tff(fact_4828_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: 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),N)) = 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),N)) ).

% add_mult_distrib2
tff(fact_4829_add__mult__distrib,axiom,
    ! [M: nat,N: 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),N)),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,N),K)) ).

% add_mult_distrib
tff(fact_4830_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: 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),N)) = 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),N)) ).

% diff_mult_distrib2
tff(fact_4831_diff__mult__distrib,axiom,
    ! [M: nat,N: 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),N)),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,N),K)) ).

% diff_mult_distrib
tff(fact_4832_nat__mult__1__right,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ).

% nat_mult_1_right
tff(fact_4833_nat__mult__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ).

% nat_mult_1
tff(fact_4834_powr__non__neg,axiom,
    ! [A2: real,X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),X)),zero_zero_real)) ).

% powr_non_neg
tff(fact_4835_powr__less__mono2__neg,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2))) ) ) ) ).

% powr_less_mono2_neg
tff(fact_4836_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y))) ).

% powr_ge_pzero
tff(fact_4837_powr__mono2,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A2))) ) ) ) ).

% powr_mono2
tff(fact_4838_powr__less__mono,axiom,
    ! [A2: real,B3: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B3))) ) ) ).

% powr_less_mono
tff(fact_4839_powr__less__cancel,axiom,
    ! [X: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B3)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) ) ) ).

% powr_less_cancel
tff(fact_4840_powr__mono,axiom,
    ! [A2: real,B3: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B3))) ) ) ).

% powr_mono
tff(fact_4841_sgn__not__eq__imp,axiom,
    ! [B3: int,A2: int] :
      ( ( aa_int_int(sgn_sgn_int,B3) != aa_int_int(sgn_sgn_int,A2) )
     => ( ( aa_int_int(sgn_sgn_int,A2) != zero_zero_int )
       => ( ( aa_int_int(sgn_sgn_int,B3) != zero_zero_int )
         => ( aa_int_int(sgn_sgn_int,A2) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,B3)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_4842_sgn__not__eq__imp,axiom,
    ! [B3: real,A2: real] :
      ( ( aa_real_real(sgn_sgn_real,B3) != aa_real_real(sgn_sgn_real,A2) )
     => ( ( aa_real_real(sgn_sgn_real,A2) != zero_zero_real )
       => ( ( aa_real_real(sgn_sgn_real,B3) != zero_zero_real )
         => ( aa_real_real(sgn_sgn_real,A2) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,B3)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_4843_sgn__not__eq__imp,axiom,
    ! [B3: rat,A2: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B3) != aa_rat_rat(sgn_sgn_rat,A2) )
     => ( ( aa_rat_rat(sgn_sgn_rat,A2) != zero_zero_rat )
       => ( ( aa_rat_rat(sgn_sgn_rat,B3) != zero_zero_rat )
         => ( aa_rat_rat(sgn_sgn_rat,A2) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(sgn_sgn_rat,B3)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_4844_sgn__not__eq__imp,axiom,
    ! [B3: code_integer,A2: code_integer] :
      ( ( sgn_sgn_Code_integer(B3) != sgn_sgn_Code_integer(A2) )
     => ( ( sgn_sgn_Code_integer(A2) != zero_z3403309356797280102nteger )
       => ( ( sgn_sgn_Code_integer(B3) != zero_z3403309356797280102nteger )
         => ( sgn_sgn_Code_integer(A2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,sgn_sgn_Code_integer(B3)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_4845_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_4846_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K))) ) ) ).

% mult_less_mono1
tff(fact_4847_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J))) ) ) ).

% mult_less_mono2
tff(fact_4848_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% Suc_mult_le_cancel1
tff(fact_4849_mult__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc
tff(fact_4850_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),I)) ) ).

% less_mult_imp_div_less
tff(fact_4851_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_4852_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)),M)) ).

% div_times_less_eq_dividend
tff(fact_4853_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))),M)) ).

% times_div_less_eq_dividend
tff(fact_4854_powr__mono2_H,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2))) ) ) ) ).

% powr_mono2'
tff(fact_4855_powr__less__mono2,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A2))) ) ) ) ).

% powr_less_mono2
tff(fact_4856_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y))) ) ) ).

% gr_one_powr
tff(fact_4857_powr__inj,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),X) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),Y) )
        <=> ( X = Y ) ) ) ) ).

% powr_inj
tff(fact_4858_ge__one__powr__ge__zero,axiom,
    ! [X: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2))) ) ) ).

% ge_one_powr_ge_zero
tff(fact_4859_powr__mono__both,axiom,
    ! [A2: real,B3: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),B3))) ) ) ) ) ).

% powr_mono_both
tff(fact_4860_powr__le1,axiom,
    ! [A2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2)),one_one_real)) ) ) ) ).

% powr_le1
tff(fact_4861_powr__divide,axiom,
    ! [X: real,Y: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),A2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A2)) ) ) ) ).

% powr_divide
tff(fact_4862_powr__mult,axiom,
    ! [X: real,Y: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A2)) ) ) ) ).

% powr_mult
tff(fact_4863_sgn__1__pos,axiom,
    ! [A2: code_integer] :
      ( ( sgn_sgn_Code_integer(A2) = one_one_Code_integer )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)) ) ).

% sgn_1_pos
tff(fact_4864_sgn__1__pos,axiom,
    ! [A2: real] :
      ( ( aa_real_real(sgn_sgn_real,A2) = one_one_real )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2)) ) ).

% sgn_1_pos
tff(fact_4865_sgn__1__pos,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A2) = one_one_rat )
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2)) ) ).

% sgn_1_pos
tff(fact_4866_sgn__1__pos,axiom,
    ! [A2: int] :
      ( ( aa_int_int(sgn_sgn_int,A2) = one_one_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2)) ) ).

% sgn_1_pos
tff(fact_4867_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) = aa_real_real(sgn_sgn_real,X) ) ) ).

% sgn_root
tff(fact_4868_abs__sgn__eq,axiom,
    ! [A2: code_integer] :
      ( ( ( A2 = zero_z3403309356797280102nteger )
       => ( abs_abs_Code_integer(sgn_sgn_Code_integer(A2)) = zero_z3403309356797280102nteger ) )
      & ( ( A2 != zero_z3403309356797280102nteger )
       => ( abs_abs_Code_integer(sgn_sgn_Code_integer(A2)) = one_one_Code_integer ) ) ) ).

% abs_sgn_eq
tff(fact_4869_abs__sgn__eq,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero_real )
       => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A2)) = zero_zero_real ) )
      & ( ( A2 != zero_zero_real )
       => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A2)) = one_one_real ) ) ) ).

% abs_sgn_eq
tff(fact_4870_abs__sgn__eq,axiom,
    ! [A2: rat] :
      ( ( ( A2 = zero_zero_rat )
       => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A2)) = zero_zero_rat ) )
      & ( ( A2 != zero_zero_rat )
       => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A2)) = one_one_rat ) ) ) ).

% abs_sgn_eq
tff(fact_4871_abs__sgn__eq,axiom,
    ! [A2: int] :
      ( ( ( A2 = zero_zero_int )
       => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A2)) = zero_zero_int ) )
      & ( ( A2 != zero_zero_int )
       => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A2)) = one_one_int ) ) ) ).

% abs_sgn_eq
tff(fact_4872_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% one_less_mult
tff(fact_4873_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% n_less_m_mult_n
tff(fact_4874_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M))) ) ) ).

% n_less_n_mult_m
tff(fact_4875_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q3)),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3))) ) ) ).

% div_less_iff_less_mult
tff(fact_4876_sgn__real__def,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero_real )
       => ( aa_real_real(sgn_sgn_real,A2) = zero_zero_real ) )
      & ( ( A2 != zero_zero_real )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
           => ( aa_real_real(sgn_sgn_real,A2) = one_one_real ) )
          & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
           => ( aa_real_real(sgn_sgn_real,A2) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).

% sgn_real_def
tff(fact_4877_sgn__1__neg,axiom,
    ! [A2: int] :
      ( ( aa_int_int(sgn_sgn_int,A2) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ).

% sgn_1_neg
tff(fact_4878_sgn__1__neg,axiom,
    ! [A2: real] :
      ( ( aa_real_real(sgn_sgn_real,A2) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ).

% sgn_1_neg
tff(fact_4879_sgn__1__neg,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A2) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ).

% sgn_1_neg
tff(fact_4880_sgn__1__neg,axiom,
    ! [A2: code_integer] :
      ( ( sgn_sgn_Code_integer(A2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger)) ) ).

% sgn_1_neg
tff(fact_4881_sgn__if,axiom,
    ! [X: int] :
      ( ( ( X = zero_zero_int )
       => ( aa_int_int(sgn_sgn_int,X) = zero_zero_int ) )
      & ( ( X != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
           => ( aa_int_int(sgn_sgn_int,X) = one_one_int ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
           => ( aa_int_int(sgn_sgn_int,X) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).

% sgn_if
tff(fact_4882_sgn__if,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( aa_real_real(sgn_sgn_real,X) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
           => ( aa_real_real(sgn_sgn_real,X) = one_one_real ) )
          & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
           => ( aa_real_real(sgn_sgn_real,X) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).

% sgn_if
tff(fact_4883_sgn__if,axiom,
    ! [X: rat] :
      ( ( ( X = zero_zero_rat )
       => ( aa_rat_rat(sgn_sgn_rat,X) = zero_zero_rat ) )
      & ( ( X != zero_zero_rat )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
           => ( aa_rat_rat(sgn_sgn_rat,X) = one_one_rat ) )
          & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
           => ( aa_rat_rat(sgn_sgn_rat,X) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ) ) ).

% sgn_if
tff(fact_4884_sgn__if,axiom,
    ! [X: code_integer] :
      ( ( ( X = zero_z3403309356797280102nteger )
       => ( sgn_sgn_Code_integer(X) = zero_z3403309356797280102nteger ) )
      & ( ( X != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X))
           => ( sgn_sgn_Code_integer(X) = one_one_Code_integer ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X))
           => ( sgn_sgn_Code_integer(X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ) ) ).

% sgn_if
tff(fact_4885_powr__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N) ) ) ).

% powr_realpow
tff(fact_4886_powr__less__iff,axiom,
    ! [B3: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B3),Y)),X))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B3),X))) ) ) ) ).

% powr_less_iff
tff(fact_4887_less__powr__iff,axiom,
    ! [B3: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B3),Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B3),X)),Y)) ) ) ) ).

% less_powr_iff
tff(fact_4888_log__less__iff,axiom,
    ! [B3: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B3),X)),Y))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B3),Y))) ) ) ) ).

% log_less_iff
tff(fact_4889_less__log__iff,axiom,
    ! [B3: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B3),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B3),Y)),X)) ) ) ) ).

% less_log_iff
tff(fact_4890_div__nat__eqI,axiom,
    ! [N: nat,Q3: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q3))))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_4891_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),Q3)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_4892_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))))) ) ).

% dividend_less_times_div
tff(fact_4893_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)))) ) ).

% dividend_less_div_times
tff(fact_4894_split__div,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I4)),J3) )
               => pp(aa_nat_bool(P,I4)) ) ) ) ) ) ).

% split_div
tff(fact_4895_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).

% mult_eq_if
tff(fact_4896_powr__neg__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X) ) ) ).

% powr_neg_one
tff(fact_4897_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).

% powr_mult_base
tff(fact_4898_sgn__power__injE,axiom,
    ! [A2: real,N: nat,X: real,B3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),N)) = X )
     => ( ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B3)),N)) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( A2 = B3 ) ) ) ) ).

% sgn_power_injE
tff(fact_4899_le__log__iff,axiom,
    ! [B3: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B3),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B3),Y)),X)) ) ) ) ).

% le_log_iff
tff(fact_4900_log__le__iff,axiom,
    ! [B3: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B3),X)),Y))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B3),Y))) ) ) ) ).

% log_le_iff
tff(fact_4901_le__powr__iff,axiom,
    ! [B3: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B3),Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B3),X)),Y)) ) ) ) ).

% le_powr_iff
tff(fact_4902_powr__le__iff,axiom,
    ! [B3: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B3),Y)),X))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B3),X))) ) ) ) ).

% powr_le_iff
tff(fact_4903_split__div_H,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
          & pp(aa_nat_bool(P,zero_zero_nat)) )
        | ? [Q6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q6)),M))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q6))))
            & pp(aa_nat_bool(P,Q6)) ) ) ) ).

% split_div'
tff(fact_4904_ln__powr__bound,axiom,
    ! [X: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2)),A2))) ) ) ).

% ln_powr_bound
tff(fact_4905_ln__powr__bound2,axiom,
    ! [X: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(ln_ln_real,X)),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),A2)),X))) ) ) ).

% ln_powr_bound2
tff(fact_4906_add__log__eq__powr,axiom,
    ! [B3: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3))
     => ( ( B3 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B3),X)) = aa_real_real(aa_rea7340605660300375871l_real(log,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B3),Y)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_4907_log__add__eq__powr,axiom,
    ! [B3: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3))
     => ( ( B3 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B3),X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B3),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_4908_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X))),N)) = X ) ) ).

% sgn_power_root
tff(fact_4909_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y)),N))) = Y ) ) ).

% root_sgn_power
tff(fact_4910_minus__log__eq__powr,axiom,
    ! [B3: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B3))
     => ( ( B3 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B3),X)) = aa_real_real(aa_rea7340605660300375871l_real(log,B3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B3),Y)),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_4911_powr__def,axiom,
    ! [X: real,A2: real] :
      ( ( ( X = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A2) = aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(ln_ln_real,X))) ) ) ) ).

% powr_def
tff(fact_4912_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_4913_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% zero_le_sgn_iff
tff(fact_4914_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sgn_sgn_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% sgn_le_0_iff
tff(fact_4915_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_4916_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_4917_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_4918_sgn__zero,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_zero
tff(fact_4919_sgn__zero,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_zero
tff(fact_4920_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_4921_mul__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(X)),some_nat(Y)) = some_nat(Z) ) ) ).

% mul_shift
tff(fact_4922_zsgn__def,axiom,
    ! [I: int] :
      ( ( ( I = zero_zero_int )
       => ( aa_int_int(sgn_sgn_int,I) = zero_zero_int ) )
      & ( ( I != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
           => ( aa_int_int(sgn_sgn_int,I) = one_one_int ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
           => ( aa_int_int(sgn_sgn_int,I) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).

% zsgn_def
tff(fact_4923_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q3: int] :
      ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L)))
       => ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),L)),R2) )
         => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),product_Pair_int_int(Q3,R2))) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_4924_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: 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),N) )
    <=> ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_4925_sgn__zero__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,X) = zero_zero_complex )
    <=> ( X = zero_zero_complex ) ) ).

% sgn_zero_iff
tff(fact_4926_sgn__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sgn_sgn_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% sgn_zero_iff
tff(fact_4927_eucl__rel__int_Osimps,axiom,
    ! [A12: int,A23: int,A32: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A12,A23),A32))
    <=> ( ? [K3: int] :
            ( ( A12 = K3 )
            & ( A23 = zero_zero_int )
            & ( A32 = product_Pair_int_int(zero_zero_int,K3) ) )
        | ? [L2: int,K3: int,Q6: int] :
            ( ( A12 = K3 )
            & ( A23 = L2 )
            & ( A32 = product_Pair_int_int(Q6,zero_zero_int) )
            & ( L2 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q6),L2) ) )
        | ? [R5: int,L2: int,K3: int,Q6: int] :
            ( ( A12 = K3 )
            & ( A23 = L2 )
            & ( A32 = product_Pair_int_int(Q6,R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L2) )
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L2)))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q6),L2)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_4928_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A32: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A12,A23),A32))
     => ( ( ( A23 = zero_zero_int )
         => ( A32 != product_Pair_int_int(zero_zero_int,A12) ) )
       => ( ! [Q4: int] :
              ( ( A32 = product_Pair_int_int(Q4,zero_zero_int) )
             => ( ( A23 != zero_zero_int )
               => ( A12 != aa_int_int(aa_int_fun_int_int(times_times_int,Q4),A23) ) ) )
         => ~ ! [R3: int,Q4: int] :
                ( ( A32 = product_Pair_int_int(Q4,R3) )
               => ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A23) )
                 => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,A23)))
                   => ( A12 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),A23)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_4929_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_4930_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
      <=> ( M = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_4931_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_4932_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_4933_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)) = 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)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_4934_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_4935_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_4936_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_4937_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N) )
      <=> ( 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)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_4938_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_4939_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_4940_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B3: real,K: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B3),X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
        <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B3),aa_nat_real(semiri5074537144036343181t_real,K))),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B3),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_4941_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero_nat) = product_Pair_int_int(one_one_int,zero_zero_int) ).

% bezw_0
tff(fact_4942_length__mul__elem,axiom,
    ! [Xs: list_list_VEBT_VEBT,N: nat] :
      ( ! [X5: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X5,set_list_VEBT_VEBT2(Xs))
         => ( size_s6755466524823107622T_VEBT(X5) = N ) )
     => ( size_s6755466524823107622T_VEBT(concat_VEBT_VEBT(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s8217280938318005548T_VEBT(Xs)),N) ) ) ).

% length_mul_elem
tff(fact_4943_length__mul__elem,axiom,
    ! [Xs: list_list_bool,N: nat] :
      ( ! [X5: list_bool] :
          ( member_list_bool(X5,set_list_bool2(Xs))
         => ( size_size_list_bool(X5) = N ) )
     => ( size_size_list_bool(concat_bool(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s9193230429364170767t_bool(Xs)),N) ) ) ).

% length_mul_elem
tff(fact_4944_length__mul__elem,axiom,
    ! [Xs: list_list_nat,N: nat] :
      ( ! [X5: list_nat] :
          ( member_list_nat(X5,set_list_nat2(Xs))
         => ( size_size_list_nat(X5) = N ) )
     => ( size_size_list_nat(concat_nat(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3023201423986296836st_nat(Xs)),N) ) ) ).

% length_mul_elem
tff(fact_4945_gbinomial__absorption_H,axiom,
    ! [K: nat,A2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_rat(gbinomial_rat(A2),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_4946_gbinomial__absorption_H,axiom,
    ! [K: nat,A2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_complex(gbinomial_complex(A2),K) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(A2),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A2),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_4947_gbinomial__absorption_H,axiom,
    ! [K: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_4948_nth__Cons__pos,axiom,
    ! [N: nat,X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( nth_VEBT_VEBT(cons_VEBT_VEBT(X,Xs),N) = nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_4949_nth__Cons__pos,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_int(nth_int(cons_int(X,Xs)),N) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_4950_nth__Cons__pos,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_4951_rotate1__length01,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),one_one_nat))
     => ( rotate1_VEBT_VEBT(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_4952_rotate1__length01,axiom,
    ! [Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Xs)),one_one_nat))
     => ( rotate1_bool(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_4953_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),one_one_nat))
     => ( rotate1_nat(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_4954_remove__def,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] : remove6466555014256735590at_nat(X,A3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) ).

% remove_def
tff(fact_4955_remove__def,axiom,
    ! [X: real,A3: set_real] : remove_real(X,A3) = minus_minus_set_real(A3,insert_real(X,bot_bot_set_real)) ).

% remove_def
tff(fact_4956_remove__def,axiom,
    ! [X: bool,A3: set_bool] : remove_bool(X,A3) = minus_minus_set_bool(A3,insert_bool(X,bot_bot_set_bool)) ).

% remove_def
tff(fact_4957_remove__def,axiom,
    ! [X: int,A3: set_int] : remove_int(X,A3) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(X,bot_bot_set_int)) ).

% remove_def
tff(fact_4958_remove__def,axiom,
    ! [X: nat,A3: set_nat] : remove_nat(X,A3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(X,bot_bot_set_nat)) ).

% remove_def
tff(fact_4959_member__remove,axiom,
    ! [X: real,Y: real,A3: set_real] :
      ( pp(aa_set_real_bool(member_real(X),remove_real(Y,A3)))
    <=> ( pp(aa_set_real_bool(member_real(X),A3))
        & ( X != Y ) ) ) ).

% member_remove
tff(fact_4960_member__remove,axiom,
    ! [X: bool,Y: bool,A3: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(X),remove_bool(Y,A3)))
    <=> ( pp(aa_set_bool_bool(member_bool(X),A3))
        & ~ ( pp(X)
          <=> pp(Y) ) ) ) ).

% member_remove
tff(fact_4961_member__remove,axiom,
    ! [X: set_nat,Y: set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),remove_set_nat(Y,A3)))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(X),A3))
        & ( X != Y ) ) ) ).

% member_remove
tff(fact_4962_member__remove,axiom,
    ! [X: nat,Y: nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),remove_nat(Y,A3)))
    <=> ( pp(aa_set_nat_bool(member_nat(X),A3))
        & ( X != Y ) ) ) ).

% member_remove
tff(fact_4963_member__remove,axiom,
    ! [X: int,Y: int,A3: set_int] :
      ( pp(aa_set_int_bool(member_int(X),remove_int(Y,A3)))
    <=> ( pp(aa_set_int_bool(member_int(X),A3))
        & ( X != Y ) ) ) ).

% member_remove
tff(fact_4964_nth__Cons__0,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] : nth_VEBT_VEBT(cons_VEBT_VEBT(X,Xs),zero_zero_nat) = X ).

% nth_Cons_0
tff(fact_4965_nth__Cons__0,axiom,
    ! [X: int,Xs: list_int] : aa_nat_int(nth_int(cons_int(X,Xs)),zero_zero_nat) = X ).

% nth_Cons_0
tff(fact_4966_nth__Cons__0,axiom,
    ! [X: nat,Xs: list_nat] : aa_nat_nat(nth_nat(cons_nat(X,Xs)),zero_zero_nat) = X ).

% nth_Cons_0
tff(fact_4967_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,zero_zero_real),aa_nat_nat(suc,K)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_4968_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_rat(gbinomial_rat(zero_zero_rat),aa_nat_nat(suc,K)) = zero_zero_rat ).

% gbinomial_0(2)
tff(fact_4969_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% gbinomial_0(2)
tff(fact_4970_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_int(gbinomial_int(zero_zero_int),aa_nat_nat(suc,K)) = zero_zero_int ).

% gbinomial_0(2)
tff(fact_4971_ceiling__zero,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,zero_zero_rat) = zero_zero_int ).

% ceiling_zero
tff(fact_4972_ceiling__zero,axiom,
    aa_real_int(archim7802044766580827645g_real,zero_zero_real) = zero_zero_int ).

% ceiling_zero
tff(fact_4973_gbinomial__0_I1_J,axiom,
    ! [A2: complex] : aa_nat_complex(gbinomial_complex(A2),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_4974_gbinomial__0_I1_J,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_4975_gbinomial__0_I1_J,axiom,
    ! [A2: rat] : aa_nat_rat(gbinomial_rat(A2),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_4976_gbinomial__0_I1_J,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A2),zero_zero_nat) = one_one_nat ).

% gbinomial_0(1)
tff(fact_4977_gbinomial__0_I1_J,axiom,
    ! [A2: int] : aa_nat_int(gbinomial_int(A2),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_4978_ceiling__le__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% ceiling_le_zero
tff(fact_4979_ceiling__le__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat)) ) ).

% ceiling_le_zero
tff(fact_4980_zero__less__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X)) ) ).

% zero_less_ceiling
tff(fact_4981_zero__less__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% zero_less_ceiling
tff(fact_4982_ceiling__less__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% ceiling_less_one
tff(fact_4983_ceiling__less__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat)) ) ).

% ceiling_less_one
tff(fact_4984_one__le__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X)) ) ).

% one_le_ceiling
tff(fact_4985_one__le__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% one_le_ceiling
tff(fact_4986_ceiling__le__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ).

% ceiling_le_one
tff(fact_4987_ceiling__le__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat)) ) ).

% ceiling_le_one
tff(fact_4988_one__less__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X)) ) ).

% one_less_ceiling
tff(fact_4989_one__less__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ).

% one_less_ceiling
tff(fact_4990_ceiling__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,one_one_real))) ) ).

% ceiling_less_zero
tff(fact_4991_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ) ).

% ceiling_less_zero
tff(fact_4992_zero__le__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)) ) ).

% zero_le_ceiling
tff(fact_4993_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X)) ) ).

% zero_le_ceiling
tff(fact_4994_ceiling__mono,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y)),aa_real_int(archim7802044766580827645g_real,X))) ) ).

% ceiling_mono
tff(fact_4995_ceiling__mono,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Y)),aa_rat_int(archim2889992004027027881ng_rat,X))) ) ).

% ceiling_mono
tff(fact_4996_ceiling__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% ceiling_less_cancel
tff(fact_4997_ceiling__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% ceiling_less_cancel
tff(fact_4998_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,N)),K) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_4999_set__subset__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),set_VEBT_VEBT2(cons_VEBT_VEBT(X,Xs))) ).

% set_subset_Cons
tff(fact_5000_set__subset__Cons,axiom,
    ! [Xs: list_nat,X: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Xs)),set_nat2(cons_nat(X,Xs)))) ).

% set_subset_Cons
tff(fact_5001_set__subset__Cons,axiom,
    ! [Xs: list_int,X: int] : pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),set_int2(cons_int(X,Xs)))) ).

% set_subset_Cons
tff(fact_5002_impossible__Cons,axiom,
    ! [Xs: list_int,Ys2: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),size_size_list_int(Ys2)))
     => ( Xs != cons_int(X,Ys2) ) ) ).

% impossible_Cons
tff(fact_5003_impossible__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys2)))
     => ( Xs != cons_VEBT_VEBT(X,Ys2) ) ) ).

% impossible_Cons
tff(fact_5004_impossible__Cons,axiom,
    ! [Xs: list_bool,Ys2: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Xs)),size_size_list_bool(Ys2)))
     => ( Xs != cons_bool(X,Ys2) ) ) ).

% impossible_Cons
tff(fact_5005_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys2: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),size_size_list_nat(Ys2)))
     => ( Xs != cons_nat(X,Ys2) ) ) ).

% impossible_Cons
tff(fact_5006_find_Osimps_I2_J,axiom,
    ! [P: fun_int_bool,X: int,Xs: list_int] :
      ( ( pp(aa_int_bool(P,X))
       => ( find_int(P,cons_int(X,Xs)) = some_int(X) ) )
      & ( ~ pp(aa_int_bool(P,X))
       => ( find_int(P,cons_int(X,Xs)) = find_int(P,Xs) ) ) ) ).

% find.simps(2)
tff(fact_5007_find_Osimps_I2_J,axiom,
    ! [P: fun_nat_bool,X: nat,Xs: list_nat] :
      ( ( pp(aa_nat_bool(P,X))
       => ( find_nat(P,cons_nat(X,Xs)) = some_nat(X) ) )
      & ( ~ pp(aa_nat_bool(P,X))
       => ( find_nat(P,cons_nat(X,Xs)) = find_nat(P,Xs) ) ) ) ).

% find.simps(2)
tff(fact_5008_find_Osimps_I2_J,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( pp(aa_Pro2232559814647311250t_bool(P,X))
       => ( find_P8199882355184865565at_nat(P,cons_P6512896166579812791at_nat(X,Xs)) = some_P7363390416028606310at_nat(X) ) )
      & ( ~ pp(aa_Pro2232559814647311250t_bool(P,X))
       => ( find_P8199882355184865565at_nat(P,cons_P6512896166579812791at_nat(X,Xs)) = find_P8199882355184865565at_nat(P,Xs) ) ) ) ).

% find.simps(2)
tff(fact_5009_find_Osimps_I2_J,axiom,
    ! [P: fun_num_bool,X: num,Xs: list_num] :
      ( ( pp(aa_num_bool(P,X))
       => ( find_num(P,cons_num(X,Xs)) = some_num(X) ) )
      & ( ~ pp(aa_num_bool(P,X))
       => ( find_num(P,cons_num(X,Xs)) = find_num(P,Xs) ) ) ) ).

% find.simps(2)
tff(fact_5010_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K)),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K))) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_5011_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(gbinomial_rat(A2),K))) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_5012_ceiling__add__le,axiom,
    ! [X: rat,Y: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y)))) ).

% ceiling_add_le
tff(fact_5013_ceiling__add__le,axiom,
    ! [X: real,Y: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y)))) ).

% ceiling_add_le
tff(fact_5014_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),size_size_list_int(Xs)))
    <=> ? [X3: int,Ys3: list_int] :
          ( ( Xs = cons_int(X3,Ys3) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_int(Ys3))) ) ) ).

% Suc_le_length_iff
tff(fact_5015_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),size_s6755466524823107622T_VEBT(Xs)))
    <=> ? [X3: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( Xs = cons_VEBT_VEBT(X3,Ys3) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s6755466524823107622T_VEBT(Ys3))) ) ) ).

% Suc_le_length_iff
tff(fact_5016_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),size_size_list_bool(Xs)))
    <=> ? [X3: bool,Ys3: list_bool] :
          ( ( Xs = cons_bool(X3,Ys3) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_bool(Ys3))) ) ) ).

% Suc_le_length_iff
tff(fact_5017_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),size_size_list_nat(Xs)))
    <=> ? [X3: nat,Ys3: list_nat] :
          ( ( Xs = cons_nat(X3,Ys3) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_nat(Ys3))) ) ) ).

% Suc_le_length_iff
tff(fact_5018_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A2),M)),aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A2),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A2),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_5019_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A2),M)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A2),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_5020_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,M)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_5021_list_Osize_I4_J,axiom,
    ! [X21: int,X222: list_int] : size_size_list_int(cons_int(X21,X222)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_int(X222)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_5022_list_Osize_I4_J,axiom,
    ! [X21: vEBT_VEBT,X222: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(cons_VEBT_VEBT(X21,X222)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_s6755466524823107622T_VEBT(X222)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_5023_list_Osize_I4_J,axiom,
    ! [X21: bool,X222: list_bool] : size_size_list_bool(cons_bool(X21,X222)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_bool(X222)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_5024_list_Osize_I4_J,axiom,
    ! [X21: nat,X222: list_nat] : size_size_list_nat(cons_nat(X21,X222)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_nat(X222)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_5025_nth__Cons_H,axiom,
    ! [N: nat,X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( nth_VEBT_VEBT(cons_VEBT_VEBT(X,Xs),N) = X ) )
      & ( ( N != zero_zero_nat )
       => ( nth_VEBT_VEBT(cons_VEBT_VEBT(X,Xs),N) = nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ) ).

% nth_Cons'
tff(fact_5026_nth__Cons_H,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(nth_int(cons_int(X,Xs)),N) = X ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(nth_int(cons_int(X,Xs)),N) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ) ).

% nth_Cons'
tff(fact_5027_nth__Cons_H,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = X ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ) ).

% nth_Cons'
tff(fact_5028_mult__ceiling__le,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A2)),aa_real_int(archim7802044766580827645g_real,B3)))) ) ) ).

% mult_ceiling_le
tff(fact_5029_mult__ceiling__le,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A2)),aa_rat_int(archim2889992004027027881ng_rat,B3)))) ) ) ).

% mult_ceiling_le
tff(fact_5030_gbinomial__reduce__nat,axiom,
    ! [K: nat,A2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_complex(gbinomial_complex(A2),K) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A2),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A2),one_one_complex)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_5031_gbinomial__reduce__nat,axiom,
    ! [K: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),one_one_real)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_5032_gbinomial__reduce__nat,axiom,
    ! [K: nat,A2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_rat(gbinomial_rat(A2),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),one_one_rat)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_5033_nth__equal__first__eq,axiom,
    ! [X: real,Xs: list_real,N: nat] :
      ( ~ pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_real(Xs)))
       => ( ( aa_nat_real(nth_real(cons_real(X,Xs)),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5034_nth__equal__first__eq,axiom,
    ! [X: set_nat,Xs: list_set_nat,N: nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s3254054031482475050et_nat(Xs)))
       => ( ( nth_set_nat(cons_set_nat(X,Xs),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5035_nth__equal__first__eq,axiom,
    ! [X: int,Xs: list_int,N: nat] :
      ( ~ pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_int(Xs)))
       => ( ( aa_nat_int(nth_int(cons_int(X,Xs)),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5036_nth__equal__first__eq,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
      ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s6755466524823107622T_VEBT(Xs)))
       => ( ( nth_VEBT_VEBT(cons_VEBT_VEBT(X,Xs),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5037_nth__equal__first__eq,axiom,
    ! [X: bool,Xs: list_bool,N: nat] :
      ( ~ pp(aa_set_bool_bool(member_bool(X),set_bool2(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_bool(Xs)))
       => ( ( pp(aa_nat_bool(nth_bool(cons_bool(X,Xs)),N))
          <=> pp(X) )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5038_nth__equal__first__eq,axiom,
    ! [X: nat,Xs: list_nat,N: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_nat(Xs)))
       => ( ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5039_nth__non__equal__first__eq,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
      ( ( X != Y )
     => ( ( nth_VEBT_VEBT(cons_VEBT_VEBT(X,Xs),N) = Y )
      <=> ( ( nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = Y )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_5040_nth__non__equal__first__eq,axiom,
    ! [X: int,Y: int,Xs: list_int,N: nat] :
      ( ( X != Y )
     => ( ( aa_nat_int(nth_int(cons_int(X,Xs)),N) = Y )
      <=> ( ( aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = Y )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_5041_nth__non__equal__first__eq,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,N: nat] :
      ( ( X != Y )
     => ( ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = Y )
      <=> ( ( aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = Y )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_5042_upto__aux__rec,axiom,
    ! [J: int,I: int,Js: list_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( upto_aux(I,J,Js) = Js ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( upto_aux(I,J,Js) = upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js)) ) ) ) ).

% upto_aux_rec
tff(fact_5043_ceiling__eq,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
       => ( aa_real_int(archim7802044766580827645g_real,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_5044_ceiling__eq,axiom,
    ! [N: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,N)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,N)),one_one_rat)))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_5045_ceiling__divide__lower,axiom,
    ! [Q3: rat,P3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P3),Q3)))),one_one_rat)),Q3)),P3)) ) ).

% ceiling_divide_lower
tff(fact_5046_ceiling__divide__lower,axiom,
    ! [Q3: real,P3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),Q3)))),one_one_real)),Q3)),P3)) ) ).

% ceiling_divide_lower
tff(fact_5047__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062treeList_H_Asummary_H_Ainfo_O_As_A_061_ANode_Ainfo_Adeg_AtreeList_H_Asummary_H_A_092_060and_062_Adeg_A_061_An_A_L_Am_A_092_060and_062_Alength_AtreeList_H_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_H_Am_A_092_060and_062_A_I_092_060forall_062t_092_060in_062set_AtreeList_H_O_Ainvar__vebt_At_An_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,Info: option4927543243414619207at_nat] :
        ~ ( ( sa = vEBT_Node(Info,deg,TreeList3,Summary3) )
          & ( deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) )
          & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
          & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),m))
          & ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList3)))
             => pp(aa_nat_bool(vEBT_invar_vebt(X4),na)) ) ) ).

% \<open>\<And>thesis. (\<And>treeList' summary' info. s = Node info deg treeList' summary' \<and> deg = n + m \<and> length treeList' = 2 ^ m \<and> invar_vebt summary' m \<and> (\<forall>t\<in>set treeList'. invar_vebt t n) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_5048_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K)))
      <=> ! [I4: int,J3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
              & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I4),J3)) ) ) ) ).

% split_pos_lemma
tff(fact_5049_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K)))
      <=> ! [I4: int,J3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
              & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I4),J3)) ) ) ) ).

% split_neg_lemma
tff(fact_5050_verit__le__mono__div__int,axiom,
    ! [A3: int,B2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A3),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),N)),aa_int_int(aa_int_fun_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),N)),zero_zero_int)),one_one_int),zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),N))) ) ) ).

% verit_le_mono_div_int
tff(fact_5051_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(71)
tff(fact_5052_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(78)
tff(fact_5053_semiring__norm_I68_J,axiom,
    ! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,one),N)) ).

% semiring_norm(68)
tff(fact_5054_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),one)) ).

% semiring_norm(75)
tff(fact_5055_case4_I10_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg))) ).

% case4(10)
tff(fact_5056_case4_I4_J,axiom,
    size_s6755466524823107622T_VEBT(treeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) ).

% case4(4)
tff(fact_5057_pow__sum,axiom,
    ! [A2: nat,B3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B3))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3) ).

% pow_sum
tff(fact_5058_a0,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ).

% a0
tff(fact_5059_case4_I7_J,axiom,
    ! [I3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m)))
     => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList2,I3)),X_1))
      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,summary2),I3)) ) ) ).

% case4(7)
tff(fact_5060_power__minus__is__div,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B3)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3)) ) ) ).

% power_minus_is_div
tff(fact_5061_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(Tree),X))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).

% member_bound
tff(fact_5062_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_5063_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_5064_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_5065_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_5066_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_5067_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_5068_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_5069_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_5070_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_5071_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_5072_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_5073_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_5074_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit0,M)),one)) ).

% semiring_norm(69)
tff(fact_5075_semiring__norm_I76_J,axiom,
    ! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),aa_num_num(bit0,N))) ).

% semiring_norm(76)
tff(fact_5076_bits__mod__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A2) = zero_zero_int ).

% bits_mod_0
tff(fact_5077_bits__mod__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A2) = zero_zero_nat ).

% bits_mod_0
tff(fact_5078_mod__self,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),A2) = zero_zero_int ).

% mod_self
tff(fact_5079_mod__self,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),A2) = zero_zero_nat ).

% mod_self
tff(fact_5080_mod__by__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),zero_zero_int) = A2 ).

% mod_by_0
tff(fact_5081_mod__by__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),zero_zero_nat) = A2 ).

% mod_by_0
tff(fact_5082_mod__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A2) = zero_zero_int ).

% mod_0
tff(fact_5083_mod__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A2) = zero_zero_nat ).

% mod_0
tff(fact_5084_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( some_nat(M) = vEBT_vebt_mint(T) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).

% misiz
tff(fact_5085_helpypredd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_pred(T,X) = some_nat(Y) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).

% helpypredd
tff(fact_5086_helpyd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_succ(T,X) = some_nat(Y) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).

% helpyd
tff(fact_5087_delt__out__of__range,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary) ) ) ) ).

% delt_out_of_range
tff(fact_5088_del__single__cont,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( X = Ma ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary) ) ) ) ).

% del_single_cont
tff(fact_5089_set__n__deg__not__0,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X5),N)) )
     => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N)) ) ) ).

% set_n_deg_not_0
tff(fact_5090_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))) ) ) ).

% mi_ma_2_deg
tff(fact_5091_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = some_nat(Ma) ) ) ) ).

% pred_max
tff(fact_5092_succ__min,axiom,
    ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = some_nat(Mi) ) ) ) ).

% succ_min
tff(fact_5093_bit__concat__def,axiom,
    ! [H: nat,L: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(H),L),D) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),D))),L) ).

% bit_concat_def
tff(fact_5094_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_5095_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_5096_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_5097_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_5098_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_5099_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_5100_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_5101_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_5102_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_5103_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_int ).

% power_zero_numeral
tff(fact_5104_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_nat ).

% power_zero_numeral
tff(fact_5105_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_real ).

% power_zero_numeral
tff(fact_5106_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_5107_mod__mult__self1__is__0,axiom,
    ! [B3: int,A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),A2)),B3) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_5108_mod__mult__self1__is__0,axiom,
    ! [B3: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),A2)),B3) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_5109_mod__mult__self2__is__0,axiom,
    ! [A2: int,B3: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3)),B3) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_5110_mod__mult__self2__is__0,axiom,
    ! [A2: nat,B3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B3)),B3) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_5111_bits__mod__by__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_5112_bits__mod__by__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),one_one_nat) = zero_zero_nat ).

% bits_mod_by_1
tff(fact_5113_mod__by__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_5114_mod__by__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),one_one_nat) = zero_zero_nat ).

% mod_by_1
tff(fact_5115_bits__mod__div__trivial,axiom,
    ! [A2: int,B3: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B3)),B3) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_5116_bits__mod__div__trivial,axiom,
    ! [A2: nat,B3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B3)),B3) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_5117_mod__div__trivial,axiom,
    ! [A2: int,B3: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B3)),B3) = zero_zero_int ).

% mod_div_trivial
tff(fact_5118_mod__div__trivial,axiom,
    ! [A2: nat,B3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B3)),B3) = zero_zero_nat ).

% mod_div_trivial
tff(fact_5119_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_5120_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_5121_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_5122_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_5123_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_5124_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_5125_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_5126_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_5127_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_5128_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z)) ) ).

% of_int_le_iff
tff(fact_5129_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z)) ) ).

% of_int_le_iff
tff(fact_5130_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z)) ) ).

% of_int_le_iff
tff(fact_5131_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ).

% of_int_less_iff
tff(fact_5132_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ).

% of_int_less_iff
tff(fact_5133_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ).

% of_int_less_iff
tff(fact_5134_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_5135_frac__of__int,axiom,
    ! [Z: int] : aa_rat_rat(archimedean_frac_rat,aa_int_rat(ring_1_of_int_rat,Z)) = zero_zero_rat ).

% frac_of_int
tff(fact_5136_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),N))
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),Ma))
          & ? [M4: nat] :
              ( ( some_nat(M4) = vEBT_vebt_mint(Summary) )
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ) ) ) ).

% mintlistlength
tff(fact_5137_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),one_one_real))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_5138_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(N)),one_on7984719198319812577d_enat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_5139_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(N)),one_one_Code_integer))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_5140_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N)),one_one_rat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_5141_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_5142_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),one_one_int))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_5143_divide__le__eq__numeral1_I1_J,axiom,
    ! [B3: real,W2: num,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),aa_num_real(numeral_numeral_real,W2))),A2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W2)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_5144_divide__le__eq__numeral1_I1_J,axiom,
    ! [B3: rat,W2: num,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),aa_num_rat(numeral_numeral_rat,W2))),A2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_5145_le__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B3: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),aa_num_real(numeral_numeral_real,W2))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W2))),B3)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_5146_le__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B3: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),aa_num_rat(numeral_numeral_rat,W2))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_num_rat(numeral_numeral_rat,W2))),B3)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_5147_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B3: rat,W2: num] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),aa_num_rat(numeral_numeral_rat,W2)) )
    <=> ( ( ( aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_num_rat(numeral_numeral_rat,W2)) = B3 ) )
        & ( ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_5148_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B3: real,W2: num] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),aa_num_real(numeral_numeral_real,W2)) )
    <=> ( ( ( aa_num_real(numeral_numeral_real,W2) != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W2)) = B3 ) )
        & ( ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_5149_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: complex,B3: complex,W2: num] :
      ( ( A2 = aa_complex_complex(divide1717551699836669952omplex(B3),numera6690914467698888265omplex(W2)) )
    <=> ( ( ( numera6690914467698888265omplex(W2) != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A2),numera6690914467698888265omplex(W2)) = B3 ) )
        & ( ( numera6690914467698888265omplex(W2) = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_5150_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B3: rat,W2: num,A2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),aa_num_rat(numeral_numeral_rat,W2)) = A2 )
    <=> ( ( ( aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat )
         => ( B3 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_num_rat(numeral_numeral_rat,W2)) ) )
        & ( ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_5151_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B3: real,W2: num,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),aa_num_real(numeral_numeral_real,W2)) = A2 )
    <=> ( ( ( aa_num_real(numeral_numeral_real,W2) != zero_zero_real )
         => ( B3 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W2)) ) )
        & ( ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_5152_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B3: complex,W2: num,A2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B3),numera6690914467698888265omplex(W2)) = A2 )
    <=> ( ( ( numera6690914467698888265omplex(W2) != zero_zero_complex )
         => ( B3 = aa_complex_complex(times_times_complex(A2),numera6690914467698888265omplex(W2)) ) )
        & ( ( numera6690914467698888265omplex(W2) = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_5153_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_5154_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_5155_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_5156_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_5157_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,one_on7984719198319812577d_enat),numera1916890842035813515d_enat(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_5158_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),numera6620942414471956472nteger(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_5159_less__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B3: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),aa_num_rat(numeral_numeral_rat,W2))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_num_rat(numeral_numeral_rat,W2))),B3)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_5160_less__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B3: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),aa_num_real(numeral_numeral_real,W2))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W2))),B3)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_5161_divide__less__eq__numeral1_I1_J,axiom,
    ! [B3: rat,W2: num,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),aa_num_rat(numeral_numeral_rat,W2))),A2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_5162_divide__less__eq__numeral1_I1_J,axiom,
    ! [B3: real,W2: num,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),aa_num_real(numeral_numeral_real,W2))),A2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W2)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_5163_mod__minus1__right,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_5164_mod__minus1__right,axiom,
    ! [A2: code_integer] : modulo364778990260209775nteger(A2,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_5165_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_le_iff
tff(fact_5166_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(N)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_le_iff
tff(fact_5167_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_le_iff
tff(fact_5168_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_le_iff
tff(fact_5169_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_le_numeral_iff
tff(fact_5170_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),numera6620942414471956472nteger(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_le_numeral_iff
tff(fact_5171_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_le_numeral_iff
tff(fact_5172_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_le_numeral_iff
tff(fact_5173_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_5174_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_5175_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_5176_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(N)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_5177_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_5178_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_5179_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_5180_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),numera6620942414471956472nteger(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_5181_ceiling__le__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_num_real(numeral_numeral_real,V))) ) ).

% ceiling_le_numeral
tff(fact_5182_ceiling__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_num_rat(numeral_numeral_rat,V))) ) ).

% ceiling_le_numeral
tff(fact_5183_numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),X)) ) ).

% numeral_less_ceiling
tff(fact_5184_numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,V)),X)) ) ).

% numeral_less_ceiling
tff(fact_5185_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_5186_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_5187_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5188_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5189_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5190_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5191_divide__le__eq__numeral1_I2_J,axiom,
    ! [B3: real,W2: num,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),A2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),B3)) ) ).

% divide_le_eq_numeral1(2)
tff(fact_5192_divide__le__eq__numeral1_I2_J,axiom,
    ! [B3: rat,W2: num,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),A2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),B3)) ) ).

% divide_le_eq_numeral1(2)
tff(fact_5193_le__divide__eq__numeral1_I2_J,axiom,
    ! [A2: real,B3: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_5194_le__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B3: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_5195_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A2: real,B3: real,W2: num] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) )
    <=> ( ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = B3 ) )
        & ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5196_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B3: rat,W2: num] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) )
    <=> ( ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = B3 ) )
        & ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5197_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A2: complex,B3: complex,W2: num] :
      ( ( A2 = aa_complex_complex(divide1717551699836669952omplex(B3),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) )
    <=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) = B3 ) )
        & ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5198_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B3: real,W2: num,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = A2 )
    <=> ( ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) != zero_zero_real )
         => ( B3 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) ) )
        & ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5199_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B3: rat,W2: num,A2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = A2 )
    <=> ( ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat )
         => ( B3 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) ) )
        & ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5200_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B3: complex,W2: num,A2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B3),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) = A2 )
    <=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) != zero_zero_complex )
         => ( B3 = aa_complex_complex(times_times_complex(A2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) ) )
        & ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5201_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5202_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,one_one_real)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5203_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5204_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5205_less__divide__eq__numeral1_I2_J,axiom,
    ! [A2: real,B3: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_5206_less__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B3: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_5207_divide__less__eq__numeral1_I2_J,axiom,
    ! [B3: real,W2: num,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),A2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),B3)) ) ).

% divide_less_eq_numeral1(2)
tff(fact_5208_divide__less__eq__numeral1_I2_J,axiom,
    ! [B3: rat,W2: num,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),A2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),B3)) ) ).

% divide_less_eq_numeral1(2)
tff(fact_5209_zero__eq__power2,axiom,
    ! [A2: rat] :
      ( ( aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_5210_zero__eq__power2,axiom,
    ! [A2: int] :
      ( ( aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_5211_zero__eq__power2,axiom,
    ! [A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat )
    <=> ( A2 = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_5212_zero__eq__power2,axiom,
    ! [A2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_5213_zero__eq__power2,axiom,
    ! [A2: complex] :
      ( ( aa_nat_complex(power_power_complex(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex )
    <=> ( A2 = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_5214_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).

% of_int_0_le_iff
tff(fact_5215_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).

% of_int_0_le_iff
tff(fact_5216_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).

% of_int_0_le_iff
tff(fact_5217_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_5218_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_5219_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_5220_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_5221_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_5222_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_5223_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% of_int_0_less_iff
tff(fact_5224_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% of_int_0_less_iff
tff(fact_5225_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% of_int_0_less_iff
tff(fact_5226_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).

% of_int_1_le_iff
tff(fact_5227_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).

% of_int_1_le_iff
tff(fact_5228_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).

% of_int_1_le_iff
tff(fact_5229_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_5230_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_5231_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_5232_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% of_int_1_less_iff
tff(fact_5233_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% of_int_1_less_iff
tff(fact_5234_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% of_int_1_less_iff
tff(fact_5235_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_5236_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_5237_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_5238_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B3: int,W2: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B3)),W2)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B3),W2)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5239_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B3: int,W2: nat,X: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B3)),W2)),aa_int_rat(ring_1_of_int_rat,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B3),W2)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5240_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B3: int,W2: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B3)),W2)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B3),W2)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5241_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B3: int,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B3)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B3),W2))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5242_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B3: int,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B3)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B3),W2))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5243_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B3: int,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B3)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B3),W2))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5244_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B3: int,W2: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B3)),W2)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B3),W2)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5245_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B3: int,W2: nat,X: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B3)),W2)),aa_int_rat(ring_1_of_int_rat,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B3),W2)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5246_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B3: int,W2: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B3)),W2)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B3),W2)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5247_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B3: int,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B3)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B3),W2))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5248_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B3: int,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B3)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B3),W2))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5249_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B3: int,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B3)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B3),W2))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5250_one__div__two__eq__zero,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% one_div_two_eq_zero
tff(fact_5251_one__div__two__eq__zero,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_5252_one__div__two__eq__zero,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% one_div_two_eq_zero
tff(fact_5253_bits__1__div__2,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% bits_1_div_2
tff(fact_5254_bits__1__div__2,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_5255_bits__1__div__2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% bits_1_div_2
tff(fact_5256_power2__eq__iff__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_5257_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_5258_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_5259_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_5260_power2__less__eq__zero__iff,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real))
    <=> ( A2 = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_5261_power2__less__eq__zero__iff,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat))
    <=> ( A2 = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_5262_power2__less__eq__zero__iff,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int))
    <=> ( A2 = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_5263_zero__less__power2,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A2 != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_5264_zero__less__power2,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A2 != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_5265_zero__less__power2,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A2 != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_5266_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_5267_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_5268_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_5269_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_5270_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_5271_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_5272_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_5273_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_5274_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_5275_ceiling__less__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real))) ) ).

% ceiling_less_numeral
tff(fact_5276_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat))) ) ).

% ceiling_less_numeral
tff(fact_5277_numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X)) ) ).

% numeral_le_ceiling
tff(fact_5278_numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)),X)) ) ).

% numeral_le_ceiling
tff(fact_5279_ceiling__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)))) ) ).

% ceiling_le_neg_numeral
tff(fact_5280_ceiling__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)))) ) ).

% ceiling_le_neg_numeral
tff(fact_5281_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),N)),aa_int_real(ring_1_of_int_real,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A2)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5282_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X)),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A2)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5283_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)),aa_int_rat(ring_1_of_int_rat,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A2)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5284_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),aa_int_int(ring_1_of_int_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A2)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5285_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5286_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A2)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5287_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A2)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5288_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5289_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),X)) ) ).

% neg_numeral_less_ceiling
tff(fact_5290_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),X)) ) ).

% neg_numeral_less_ceiling
tff(fact_5291_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)),aa_int_rat(ring_1_of_int_rat,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A2)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5292_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),N)),aa_int_real(ring_1_of_int_real,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A2)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5293_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),aa_int_int(ring_1_of_int_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A2)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5294_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X)),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A2)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5295_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A2)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5296_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5297_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5298_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A2)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5299_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5300_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5301_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5302_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5303_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5304_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5305_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5306_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)),aa_nat_Code_integer(semiri4939895301339042750nteger,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5307_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5308_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),N)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5309_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)),aa_nat_Code_integer(semiri4939895301339042750nteger,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5310_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),N)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5311_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5312_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5313_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5314_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5315_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5316_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5317_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5318_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5319_ceiling__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real))) ) ).

% ceiling_less_neg_numeral
tff(fact_5320_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat))) ) ).

% ceiling_less_neg_numeral
tff(fact_5321_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),X)) ) ).

% neg_numeral_le_ceiling
tff(fact_5322_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X)) ) ).

% neg_numeral_le_ceiling
tff(fact_5323_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5324_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A2)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5325_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A2)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5326_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A2)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5327_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N)),aa_int_real(ring_1_of_int_real,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A2)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5328_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A2)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5329_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)),aa_int_rat(ring_1_of_int_rat,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A2)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5330_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),aa_int_int(ring_1_of_int_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A2)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5331_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A2)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5332_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5333_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A2)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5334_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A2)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5335_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),aa_int_int(ring_1_of_int_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A2)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5336_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N)),aa_int_real(ring_1_of_int_real,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A2)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5337_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)),aa_int_rat(ring_1_of_int_rat,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A2)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5338_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A2)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5339_sprop1,axiom,
    ( ( sa = vEBT_Node(info,deg,treeList,summary) )
    & ( deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) )
    & ( size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
    & pp(aa_nat_bool(vEBT_invar_vebt(summary),m))
    & ! [X4: vEBT_VEBT] :
        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(treeList)))
       => pp(aa_nat_bool(vEBT_invar_vebt(X4),na)) ) ) ).

% sprop1
tff(fact_5340_divmod__digit__0_I2_J,axiom,
    ! [B3: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B3))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B3))),B3))
       => ( modulo364778990260209775nteger(A2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B3)) = modulo364778990260209775nteger(A2,B3) ) ) ) ).

% divmod_digit_0(2)
tff(fact_5341_divmod__digit__0_I2_J,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3))),B3))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B3) ) ) ) ).

% divmod_digit_0(2)
tff(fact_5342_divmod__digit__0_I2_J,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3))),B3))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B3) ) ) ) ).

% divmod_digit_0(2)
tff(fact_5343_pos2,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% pos2
tff(fact_5344_zero__power2,axiom,
    aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat ).

% zero_power2
tff(fact_5345_zero__power2,axiom,
    aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int ).

% zero_power2
tff(fact_5346_zero__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% zero_power2
tff(fact_5347_zero__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real ).

% zero_power2
tff(fact_5348_zero__power2,axiom,
    aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex ).

% zero_power2
tff(fact_5349_numeral__2__eq__2,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% numeral_2_eq_2
tff(fact_5350_less__exp,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% less_exp
tff(fact_5351_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M))) ) ).

% self_le_ge2_pow
tff(fact_5352_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% power2_nat_le_eq_le
tff(fact_5353_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% power2_nat_le_imp_le
tff(fact_5354_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_5355_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_5356_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_5357_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_5358_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_5359_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_5360_card__2__iff_H,axiom,
    ! [S: set_complex] :
      ( ( finite_card_complex(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),S))
          & ? [Xa3: complex] :
              ( pp(aa_set_complex_bool(member_complex(Xa3),S))
              & ( X3 != Xa3 )
              & ! [Xb2: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xb2),S))
                 => ( ( Xb2 = X3 )
                    | ( Xb2 = Xa3 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_5361_card__2__iff_H,axiom,
    ! [S: set_list_nat] :
      ( ( finite_card_list_nat(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X3: list_nat] :
          ( member_list_nat(X3,S)
          & ? [Xa3: list_nat] :
              ( member_list_nat(Xa3,S)
              & ( X3 != Xa3 )
              & ! [Xb2: list_nat] :
                  ( member_list_nat(Xb2,S)
                 => ( ( Xb2 = X3 )
                    | ( Xb2 = Xa3 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_5362_card__2__iff_H,axiom,
    ! [S: set_set_nat] :
      ( ( finite_card_set_nat(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X3: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X3),S))
          & ? [Xa3: set_nat] :
              ( pp(aa_set_set_nat_bool(member_set_nat(Xa3),S))
              & ( X3 != Xa3 )
              & ! [Xb2: set_nat] :
                  ( pp(aa_set_set_nat_bool(member_set_nat(Xb2),S))
                 => ( ( Xb2 = X3 )
                    | ( Xb2 = Xa3 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_5363_card__2__iff_H,axiom,
    ! [S: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),S))
          & ? [Xa3: nat] :
              ( pp(aa_set_nat_bool(member_nat(Xa3),S))
              & ( X3 != Xa3 )
              & ! [Xb2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xb2),S))
                 => ( ( Xb2 = X3 )
                    | ( Xb2 = Xa3 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_5364_card__2__iff_H,axiom,
    ! [S: set_int] :
      ( ( finite_card_int(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),S))
          & ? [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),S))
              & ( X3 != Xa3 )
              & ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),S))
                 => ( ( Xb2 = X3 )
                    | ( Xb2 = Xa3 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_5365_le__num__One__iff,axiom,
    ! [X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),one))
    <=> ( X = one ) ) ).

% le_num_One_iff
tff(fact_5366_bits__stable__imp__add__self,axiom,
    ! [A2: code_integer] :
      ( ( divide6298287555418463151nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A2 )
     => ( plus_p5714425477246183910nteger(A2,modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_5367_bits__stable__imp__add__self,axiom,
    ! [A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A2 )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_5368_bits__stable__imp__add__self,axiom,
    ! [A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A2 )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_5369_divmod__digit__0_I1_J,axiom,
    ! [B3: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B3))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B3))),B3))
       => ( times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),divide6298287555418463151nteger(A2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B3))) = divide6298287555418463151nteger(A2,B3) ) ) ) ).

% divmod_digit_0(1)
tff(fact_5370_divmod__digit__0_I1_J,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3))),B3))
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3) ) ) ) ).

% divmod_digit_0(1)
tff(fact_5371_divmod__digit__0_I1_J,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3))),B3))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B3) ) ) ) ).

% divmod_digit_0(1)
tff(fact_5372_sum__squares__bound,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_squares_bound
tff(fact_5373_sum__squares__bound,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_squares_bound
tff(fact_5374_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( modulo364778990260209775nteger(times_3573771949741848930nteger(A2,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = times_3573771949741848930nteger(modulo364778990260209775nteger(A2,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5375_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5376_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5377_half__gt__zero,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ).

% half_gt_zero
tff(fact_5378_half__gt__zero,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% half_gt_zero
tff(fact_5379_half__gt__zero__iff,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A2),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2)) ) ).

% half_gt_zero_iff
tff(fact_5380_half__gt__zero__iff,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2)) ) ).

% half_gt_zero_iff
tff(fact_5381_field__less__half__sum,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_5382_field__less__half__sum,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_5383_power2__le__imp__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_5384_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_5385_power2__le__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_5386_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_5387_power2__eq__imp__eq,axiom,
    ! [X: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_5388_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_5389_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_5390_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_5391_zero__le__power2,axiom,
    ! [A2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_5392_zero__le__power2,axiom,
    ! [A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_5393_zero__le__power2,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_5394_power2__less__0,axiom,
    ! [A2: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)) ).

% power2_less_0
tff(fact_5395_power2__less__0,axiom,
    ! [A2: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)) ).

% power2_less_0
tff(fact_5396_power2__less__0,axiom,
    ! [A2: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)) ).

% power2_less_0
tff(fact_5397_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_5398_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_5399_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_5400_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_5401_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_5402_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_5403_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
tff(fact_5404_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_5405_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_5406_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
tff(fact_5407_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_5408_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_5409_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_5410_abs__le__square__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),abs_abs_Code_integer(Y)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_5411_abs__le__square__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_5412_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),aa_rat_rat(abs_abs_rat,Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_5413_abs__le__square__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),aa_int_int(abs_abs_int,Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_5414_less__2__cases,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_5415_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_5416_card__2__iff,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X3: product_prod_nat_nat,Y2: product_prod_nat_nat] :
          ( ( S = insert8211810215607154385at_nat(X3,insert8211810215607154385at_nat(Y2,bot_bo2099793752762293965at_nat)) )
          & ( X3 != Y2 ) ) ) ).

% card_2_iff
tff(fact_5417_card__2__iff,axiom,
    ! [S: set_complex] :
      ( ( finite_card_complex(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X3: complex,Y2: complex] :
          ( ( S = insert_complex(X3,insert_complex(Y2,bot_bot_set_complex)) )
          & ( X3 != Y2 ) ) ) ).

% card_2_iff
tff(fact_5418_card__2__iff,axiom,
    ! [S: set_list_nat] :
      ( ( finite_card_list_nat(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X3: list_nat,Y2: list_nat] :
          ( ( S = insert_list_nat(X3,insert_list_nat(Y2,bot_bot_set_list_nat)) )
          & ( X3 != Y2 ) ) ) ).

% card_2_iff
tff(fact_5419_card__2__iff,axiom,
    ! [S: set_set_nat] :
      ( ( finite_card_set_nat(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X3: set_nat,Y2: set_nat] :
          ( ( S = insert_set_nat(X3,insert_set_nat(Y2,bot_bot_set_set_nat)) )
          & ( X3 != Y2 ) ) ) ).

% card_2_iff
tff(fact_5420_card__2__iff,axiom,
    ! [S: set_real] :
      ( ( finite_card_real(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X3: real,Y2: real] :
          ( ( S = insert_real(X3,insert_real(Y2,bot_bot_set_real)) )
          & ( X3 != Y2 ) ) ) ).

% card_2_iff
tff(fact_5421_card__2__iff,axiom,
    ! [S: set_bool] :
      ( ( finite_card_bool(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X3: bool,Y2: bool] :
          ( ( S = insert_bool(X3,insert_bool(Y2,bot_bot_set_bool)) )
          & ~ ( pp(X3)
            <=> pp(Y2) ) ) ) ).

% card_2_iff
tff(fact_5422_card__2__iff,axiom,
    ! [S: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X3: nat,Y2: nat] :
          ( ( S = insert_nat(X3,insert_nat(Y2,bot_bot_set_nat)) )
          & ( X3 != Y2 ) ) ) ).

% card_2_iff
tff(fact_5423_card__2__iff,axiom,
    ! [S: set_int] :
      ( ( finite_card_int(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X3: int,Y2: int] :
          ( ( S = insert_int(X3,insert_int(Y2,bot_bot_set_int)) )
          & ( X3 != Y2 ) ) ) ).

% card_2_iff
tff(fact_5424_nat__induct2,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( pp(aa_nat_bool(P,one_one_nat))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(P,N2))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct2
tff(fact_5425_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),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),N)))) ) ).

% diff_le_diff_pow
tff(fact_5426_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% realpow_square_minus_le
tff(fact_5427_numeral__1__eq__Suc__0,axiom,
    aa_num_nat(numeral_numeral_nat,one) = aa_nat_nat(suc,zero_zero_nat) ).

% numeral_1_eq_Suc_0
tff(fact_5428_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z3: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z3))) ).

% ex_le_of_int
tff(fact_5429_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z3: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z3))) ).

% ex_le_of_int
tff(fact_5430_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z3: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z3))) ).

% ex_less_of_int
tff(fact_5431_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z3: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z3))) ).

% ex_less_of_int
tff(fact_5432_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z3: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z3)),X)) ).

% ex_of_int_less
tff(fact_5433_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z3: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z3)),X)) ).

% ex_of_int_less
tff(fact_5434_mod__double__modulus,axiom,
    ! [M: code_integer,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),M))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
       => ( ( modulo364778990260209775nteger(X,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M)) = modulo364778990260209775nteger(X,M) )
          | ( modulo364778990260209775nteger(X,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(X,M),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5435_mod__double__modulus,axiom,
    ! [M: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5436_mod__double__modulus,axiom,
    ! [M: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),M) )
          | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5437_divmod__digit__1_I2_J,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B3))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B3),modulo364778990260209775nteger(A2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B3))))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B3)),B3) = modulo364778990260209775nteger(A2,B3) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5438_divmod__digit__1_I2_J,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3))))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3))),B3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B3) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5439_divmod__digit__1_I2_J,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3))))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3))),B3) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B3) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5440_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,N) ).

% zero_neq_numeral
tff(fact_5441_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,N) ).

% zero_neq_numeral
tff(fact_5442_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_num_real(numeral_numeral_real,N) ).

% zero_neq_numeral
tff(fact_5443_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_num_int(numeral_numeral_int,N) ).

% zero_neq_numeral
tff(fact_5444_zero__neq__numeral,axiom,
    ! [N: num] : zero_z5237406670263579293d_enat != numera1916890842035813515d_enat(N) ).

% zero_neq_numeral
tff(fact_5445_zero__neq__numeral,axiom,
    ! [N: num] : zero_z3403309356797280102nteger != numera6620942414471956472nteger(N) ).

% zero_neq_numeral
tff(fact_5446_power2__less__imp__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_5447_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_5448_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_5449_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_5450_sum__power2__ge__zero,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_5451_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_5452_sum__power2__ge__zero,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_5453_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real))
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_5454_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat))
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_5455_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int))
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_5456_sum__power2__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_5457_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_5458_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_5459_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)) ).

% not_sum_power2_lt_zero
tff(fact_5460_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)) ).

% not_sum_power2_lt_zero
tff(fact_5461_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)) ).

% not_sum_power2_lt_zero
tff(fact_5462_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_5463_square__le__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% square_le_1
tff(fact_5464_square__le__1,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)) ) ) ).

% square_le_1
tff(fact_5465_square__le__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)) ) ) ).

% square_le_1
tff(fact_5466_square__le__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)) ) ) ).

% square_le_1
tff(fact_5467_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_5468_power2__le__iff__abs__le,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_5469_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_5470_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_5471_zero__le__even__power_H,axiom,
    ! [A2: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% zero_le_even_power'
tff(fact_5472_zero__le__even__power_H,axiom,
    ! [A2: rat,N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% zero_le_even_power'
tff(fact_5473_zero__le__even__power_H,axiom,
    ! [A2: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% zero_le_even_power'
tff(fact_5474_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),one_one_Code_integer)) ) ).

% abs_square_le_1
tff(fact_5475_abs__square__le__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)) ) ).

% abs_square_le_1
tff(fact_5476_abs__square__le__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat)) ) ).

% abs_square_le_1
tff(fact_5477_abs__square__le__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),one_one_int)) ) ).

% abs_square_le_1
tff(fact_5478_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(X)),one_one_Code_integer)) ) ).

% abs_square_less_1
tff(fact_5479_abs__square__less__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)) ) ).

% abs_square_less_1
tff(fact_5480_abs__square__less__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat)) ) ).

% abs_square_less_1
tff(fact_5481_abs__square__less__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int)) ) ).

% abs_square_less_1
tff(fact_5482_nat__bit__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,N2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ) )
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(P,N2))
             => pp(aa_nat_bool(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_bit_induct
tff(fact_5483_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% div_2_gt_zero
tff(fact_5484_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_5485_arith__geo__mean,axiom,
    ! [U: real,X: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_5486_arith__geo__mean,axiom,
    ! [U: rat,X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,U),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_5487_divmod__digit__1_I1_J,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B3))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B3),modulo364778990260209775nteger(A2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B3))))
         => ( plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),divide6298287555418463151nteger(A2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B3))),one_one_Code_integer) = divide6298287555418463151nteger(A2,B3) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_5488_divmod__digit__1_I1_J,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3))))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B3) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_5489_divmod__digit__1_I1_J,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3))))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_5490_odd__0__le__power__imp__0__le,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_5491_odd__0__le__power__imp__0__le,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_5492_odd__0__le__power__imp__0__le,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_5493_odd__power__less__zero,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_real)) ) ).

% odd_power_less_zero
tff(fact_5494_odd__power__less__zero,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_rat)) ) ).

% odd_power_less_zero
tff(fact_5495_odd__power__less__zero,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_int)) ) ).

% odd_power_less_zero
tff(fact_5496_ex__power__ivl2,axiom,
    ! [B3: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
       => ? [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),N2)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_5497_ex__power__ivl1,axiom,
    ! [B3: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),K))
       => ? [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),N2)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_5498_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B3)),A2)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_5499_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B3)),A2)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_5500_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B3)),B3)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_5501_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B3)),B3)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_5502_mod__eq__self__iff__div__eq__0,axiom,
    ! [A2: int,B3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B3) = A2 )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_5503_mod__eq__self__iff__div__eq__0,axiom,
    ! [A2: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B3) = A2 )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B3) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_5504_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,N))) ).

% zero_le_numeral
tff(fact_5505_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(N))) ).

% zero_le_numeral
tff(fact_5506_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),numera6620942414471956472nteger(N))) ).

% zero_le_numeral
tff(fact_5507_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% zero_le_numeral
tff(fact_5508_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% zero_le_numeral
tff(fact_5509_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,N))) ).

% zero_le_numeral
tff(fact_5510_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),zero_zero_real)) ).

% not_numeral_le_zero
tff(fact_5511_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(N)),zero_z5237406670263579293d_enat)) ).

% not_numeral_le_zero
tff(fact_5512_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(N)),zero_z3403309356797280102nteger)) ).

% not_numeral_le_zero
tff(fact_5513_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N)),zero_zero_rat)) ).

% not_numeral_le_zero
tff(fact_5514_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).

% not_numeral_le_zero
tff(fact_5515_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int)) ).

% not_numeral_le_zero
tff(fact_5516_exp__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% exp_bound
tff(fact_5517_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),zero_zero_rat)) ).

% not_numeral_less_zero
tff(fact_5518_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).

% not_numeral_less_zero
tff(fact_5519_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N)),zero_zero_real)) ).

% not_numeral_less_zero
tff(fact_5520_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int)) ).

% not_numeral_less_zero
tff(fact_5521_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(N)),zero_z5237406670263579293d_enat)) ).

% not_numeral_less_zero
tff(fact_5522_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(N)),zero_z3403309356797280102nteger)) ).

% not_numeral_less_zero
tff(fact_5523_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% zero_less_numeral
tff(fact_5524_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% zero_less_numeral
tff(fact_5525_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,N))) ).

% zero_less_numeral
tff(fact_5526_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,N))) ).

% zero_less_numeral
tff(fact_5527_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(N))) ).

% zero_less_numeral
tff(fact_5528_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),numera6620942414471956472nteger(N))) ).

% zero_less_numeral
tff(fact_5529_le__of__int__ceiling,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)))) ).

% le_of_int_ceiling
tff(fact_5530_le__of__int__ceiling,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X)))) ).

% le_of_int_ceiling
tff(fact_5531_one__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,N))) ).

% one_le_numeral
tff(fact_5532_one__le__numeral,axiom,
    ! [N: num] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,one_on7984719198319812577d_enat),numera1916890842035813515d_enat(N))) ).

% one_le_numeral
tff(fact_5533_one__le__numeral,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),numera6620942414471956472nteger(N))) ).

% one_le_numeral
tff(fact_5534_one__le__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% one_le_numeral
tff(fact_5535_one__le__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% one_le_numeral
tff(fact_5536_one__le__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,N))) ).

% one_le_numeral
tff(fact_5537_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),one_one_rat)) ).

% not_numeral_less_one
tff(fact_5538_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat)) ).

% not_numeral_less_one
tff(fact_5539_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N)),one_one_real)) ).

% not_numeral_less_one
tff(fact_5540_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),one_one_int)) ).

% not_numeral_less_one
tff(fact_5541_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(N)),one_on7984719198319812577d_enat)) ).

% not_numeral_less_one
tff(fact_5542_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(N)),one_one_Code_integer)) ).

% not_numeral_less_one
tff(fact_5543_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,N))) ).

% neg_numeral_le_numeral
tff(fact_5544_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N))) ).

% neg_numeral_le_numeral
tff(fact_5545_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N))) ).

% neg_numeral_le_numeral
tff(fact_5546_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N))) ).

% neg_numeral_le_numeral
tff(fact_5547_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_5548_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).

% not_numeral_le_neg_numeral
tff(fact_5549_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_5550_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_5551_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% zero_neq_neg_numeral
tff(fact_5552_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) ).

% zero_neq_neg_numeral
tff(fact_5553_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) ).

% zero_neq_neg_numeral
tff(fact_5554_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) ).

% zero_neq_neg_numeral
tff(fact_5555_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% zero_neq_neg_numeral
tff(fact_5556_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),K)),M)) ) ).

% zmod_le_nonneg_dividend
tff(fact_5557_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5558_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5559_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5560_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5561_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N))) ).

% neg_numeral_less_numeral
tff(fact_5562_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,N))) ).

% neg_numeral_less_numeral
tff(fact_5563_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N))) ).

% neg_numeral_less_numeral
tff(fact_5564_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N))) ).

% neg_numeral_less_numeral
tff(fact_5565_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),L)) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_5566_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ) ).

% neg_mod_bound
tff(fact_5567_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_5568_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X5),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12))
               => ( ! [X5: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                     => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_12)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_5569_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_5570_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B3))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_5571_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B3))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_5572_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B3) = A2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_5573_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B3) = A2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_5574_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X5),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12))
               => ( ! [X5: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                     => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_12)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_5575_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))),zero_zero_real)) ).

% neg_numeral_le_zero
tff(fact_5576_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))),zero_z3403309356797280102nteger)) ).

% neg_numeral_le_zero
tff(fact_5577_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))),zero_zero_rat)) ).

% neg_numeral_le_zero
tff(fact_5578_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int)) ).

% neg_numeral_le_zero
tff(fact_5579_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_zero_le_neg_numeral
tff(fact_5580_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).

% not_zero_le_neg_numeral
tff(fact_5581_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_zero_le_neg_numeral
tff(fact_5582_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_zero_le_neg_numeral
tff(fact_5583_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_zero_less_neg_numeral
tff(fact_5584_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_zero_less_neg_numeral
tff(fact_5585_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_zero_less_neg_numeral
tff(fact_5586_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).

% not_zero_less_neg_numeral
tff(fact_5587_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int)) ).

% neg_numeral_less_zero
tff(fact_5588_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))),zero_zero_real)) ).

% neg_numeral_less_zero
tff(fact_5589_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))),zero_zero_rat)) ).

% neg_numeral_less_zero
tff(fact_5590_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))),zero_z3403309356797280102nteger)) ).

% neg_numeral_less_zero
tff(fact_5591_ceiling__le,axiom,
    ! [X: real,A2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A2)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),A2)) ) ).

% ceiling_le
tff(fact_5592_ceiling__le,axiom,
    ! [X: rat,A2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A2)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),A2)) ) ).

% ceiling_le
tff(fact_5593_ceiling__le__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z))) ) ).

% ceiling_le_iff
tff(fact_5594_ceiling__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% ceiling_le_iff
tff(fact_5595_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B3: rat,C: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C) = B3 ) )
        & ( ( C = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_5596_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B3: real,C: real] :
      ( ( aa_num_real(numeral_numeral_real,W2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C) = B3 ) )
        & ( ( C = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_5597_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B3: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W2) = aa_complex_complex(divide1717551699836669952omplex(B3),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C) = B3 ) )
        & ( ( C = zero_zero_complex )
         => ( numera6690914467698888265omplex(W2) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_5598_divide__eq__eq__numeral_I1_J,axiom,
    ! [B3: rat,C: rat,W2: num] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C) = aa_num_rat(numeral_numeral_rat,W2) )
    <=> ( ( ( C != zero_zero_rat )
         => ( B3 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C) ) )
        & ( ( C = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_5599_divide__eq__eq__numeral_I1_J,axiom,
    ! [B3: real,C: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C) = aa_num_real(numeral_numeral_real,W2) )
    <=> ( ( ( C != zero_zero_real )
         => ( B3 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C) ) )
        & ( ( C = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_5600_divide__eq__eq__numeral_I1_J,axiom,
    ! [B3: complex,C: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B3),C) = numera6690914467698888265omplex(W2) )
    <=> ( ( ( C != zero_zero_complex )
         => ( B3 = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C) ) )
        & ( ( C = zero_zero_complex )
         => ( numera6690914467698888265omplex(W2) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_5601_less__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)) ) ).

% less_ceiling_iff
tff(fact_5602_less__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),X)) ) ).

% less_ceiling_iff
tff(fact_5603_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)))) ).

% not_one_le_neg_numeral
tff(fact_5604_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).

% not_one_le_neg_numeral
tff(fact_5605_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)))) ).

% not_one_le_neg_numeral
tff(fact_5606_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).

% not_one_le_neg_numeral
tff(fact_5607_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_le_neg_one
tff(fact_5608_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% not_numeral_le_neg_one
tff(fact_5609_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% not_numeral_le_neg_one
tff(fact_5610_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int))) ).

% not_numeral_le_neg_one
tff(fact_5611_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,one_one_real))) ).

% neg_numeral_le_neg_one
tff(fact_5612_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% neg_numeral_le_neg_one
tff(fact_5613_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% neg_numeral_le_neg_one
tff(fact_5614_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int))) ).

% neg_numeral_le_neg_one
tff(fact_5615_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,M))) ).

% neg_one_le_numeral
tff(fact_5616_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M))) ).

% neg_one_le_numeral
tff(fact_5617_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,M))) ).

% neg_one_le_numeral
tff(fact_5618_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M))) ).

% neg_one_le_numeral
tff(fact_5619_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real)) ).

% neg_numeral_le_one
tff(fact_5620_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),one_one_Code_integer)) ).

% neg_numeral_le_one
tff(fact_5621_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat)) ).

% neg_numeral_le_one
tff(fact_5622_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int)) ).

% neg_numeral_le_one
tff(fact_5623_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int)) ).

% neg_numeral_less_one
tff(fact_5624_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real)) ).

% neg_numeral_less_one
tff(fact_5625_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat)) ).

% neg_numeral_less_one
tff(fact_5626_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),one_one_Code_integer)) ).

% neg_numeral_less_one
tff(fact_5627_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M))) ).

% neg_one_less_numeral
tff(fact_5628_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,M))) ).

% neg_one_less_numeral
tff(fact_5629_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,M))) ).

% neg_one_less_numeral
tff(fact_5630_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M))) ).

% neg_one_less_numeral
tff(fact_5631_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int))) ).

% not_numeral_less_neg_one
tff(fact_5632_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_less_neg_one
tff(fact_5633_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% not_numeral_less_neg_one
tff(fact_5634_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% not_numeral_less_neg_one
tff(fact_5635_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).

% not_one_less_neg_numeral
tff(fact_5636_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)))) ).

% not_one_less_neg_numeral
tff(fact_5637_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)))) ).

% not_one_less_neg_numeral
tff(fact_5638_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).

% not_one_less_neg_numeral
tff(fact_5639_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5640_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5641_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5642_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5643_neg__mod__conj,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B3)),zero_zero_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B3))) ) ) ).

% neg_mod_conj
tff(fact_5644_pos__mod__conj,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B3)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B3)),B3)) ) ) ).

% pos_mod_conj
tff(fact_5645_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I),K) = I )
    <=> ( ( K = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).

% zmod_trivial_iff
tff(fact_5646_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),zero_zero_int)) ) ).

% neg_mod_sign
tff(fact_5647_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_5648_real__of__int__div4,axiom,
    ! [N: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X)))) ).

% real_of_int_div4
tff(fact_5649_zdiv__mono__strict,axiom,
    ! [A3: int,B2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A3),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),N) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),N) = zero_zero_int )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),N))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_5650_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))),aa_int_int(abs_abs_int,L))) ) ).

% abs_mod_less
tff(fact_5651_num_Osize_I5_J,axiom,
    ! [X2: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_5652_of__int__nonneg,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))) ) ).

% of_int_nonneg
tff(fact_5653_of__int__nonneg,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% of_int_nonneg
tff(fact_5654_of__int__nonneg,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))) ) ).

% of_int_nonneg
tff(fact_5655_of__int__leD,axiom,
    ! [N: int,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),X)) ) ) ).

% of_int_leD
tff(fact_5656_of__int__leD,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% of_int_leD
tff(fact_5657_of__int__leD,axiom,
    ! [N: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ) ).

% of_int_leD
tff(fact_5658_of__int__leD,axiom,
    ! [N: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),X)) ) ) ).

% of_int_leD
tff(fact_5659_of__int__pos,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))) ) ).

% of_int_pos
tff(fact_5660_of__int__pos,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% of_int_pos
tff(fact_5661_of__int__pos,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))) ) ).

% of_int_pos
tff(fact_5662_of__int__lessD,axiom,
    ! [N: int,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),X)) ) ) ).

% of_int_lessD
tff(fact_5663_of__int__lessD,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% of_int_lessD
tff(fact_5664_of__int__lessD,axiom,
    ! [N: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X)) ) ) ).

% of_int_lessD
tff(fact_5665_of__int__lessD,axiom,
    ! [N: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),X)) ) ) ).

% of_int_lessD
tff(fact_5666_floor__exists1,axiom,
    ! [X: real] :
    ? [X5: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X5)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),one_one_int))))
      & ! [Y4: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y4)),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int)))) )
         => ( Y4 = X5 ) ) ) ).

% floor_exists1
tff(fact_5667_floor__exists1,axiom,
    ! [X: rat] :
    ? [X5: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X5)),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),one_one_int))))
      & ! [Y4: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y4)),X))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int)))) )
         => ( Y4 = X5 ) ) ) ).

% floor_exists1
tff(fact_5668_floor__exists,axiom,
    ! [X: real] :
    ? [Z3: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z3)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z3),one_one_int)))) ) ).

% floor_exists
tff(fact_5669_floor__exists,axiom,
    ! [X: rat] :
    ? [Z3: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z3)),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z3),one_one_int)))) ) ).

% floor_exists
tff(fact_5670_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5671_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,R2),one_one_rat))) ).

% of_int_ceiling_le_add_one
tff(fact_5672_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5673_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),one_one_rat)),R2)) ).

% of_int_ceiling_diff_one_le
tff(fact_5674_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_int_rat(ring_1_of_int_rat,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_5675_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_5676_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_5677_ceiling__log__nat__eq__if,axiom,
    ! [B3: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3))
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B3)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_5678_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C)),B3)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_5679_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),B3)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W2)),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_5680_divide__less__eq__numeral_I1_J,axiom,
    ! [B3: rat,C: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)),aa_num_rat(numeral_numeral_rat,W2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C)),B3)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_5681_divide__less__eq__numeral_I1_J,axiom,
    ! [B3: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)),aa_num_real(numeral_numeral_real,W2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),B3)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W2))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_5682_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B3: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C) = B3 ) )
        & ( ( C = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5683_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B3: rat,C: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C) = B3 ) )
        & ( ( C = zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5684_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B3: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = aa_complex_complex(divide1717551699836669952omplex(B3),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C) = B3 ) )
        & ( ( C = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5685_divide__eq__eq__numeral_I2_J,axiom,
    ! [B3: real,C: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) )
    <=> ( ( ( C != zero_zero_real )
         => ( B3 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C) ) )
        & ( ( C = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5686_divide__eq__eq__numeral_I2_J,axiom,
    ! [B3: rat,C: rat,W2: num] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) )
    <=> ( ( ( C != zero_zero_rat )
         => ( B3 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C) ) )
        & ( ( C = zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5687_divide__eq__eq__numeral_I2_J,axiom,
    ! [B3: complex,C: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B3),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) )
    <=> ( ( ( C != zero_zero_complex )
         => ( B3 = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C) ) )
        & ( ( C = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5688_int__le__real__less,axiom,
    ! [N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),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_5689_int__less__real__le,axiom,
    ! [N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)),aa_int_real(ring_1_of_int_real,M))) ) ).

% int_less_real_le
tff(fact_5690_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_5691_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_5692_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B3: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B3)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) )
        <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),N)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_5693_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B3)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B3)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5694_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B3)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5695_ceiling__correct,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,X))),one_one_real)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)))) ) ).

% ceiling_correct
tff(fact_5696_ceiling__correct,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X))),one_one_rat)),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X)))) ) ).

% ceiling_correct
tff(fact_5697_ceiling__unique,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z)))
       => ( aa_real_int(archim7802044766580827645g_real,X) = Z ) ) ) ).

% ceiling_unique
tff(fact_5698_ceiling__unique,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z)))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = Z ) ) ) ).

% ceiling_unique
tff(fact_5699_ceiling__eq__iff,axiom,
    ! [X: real,A2: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,X) = A2 )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,A2)),one_one_real)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A2))) ) ) ).

% ceiling_eq_iff
tff(fact_5700_ceiling__eq__iff,axiom,
    ! [X: rat,A2: int] :
      ( ( aa_rat_int(archim2889992004027027881ng_rat,X) = A2 )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,A2)),one_one_rat)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A2))) ) ) ).

% ceiling_eq_iff
tff(fact_5701_ceiling__split,axiom,
    ! [P: fun_int_bool,T: real] :
      ( pp(aa_int_bool(P,aa_real_int(archim7802044766580827645g_real,T)))
    <=> ! [I4: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,I4)),one_one_real)),T))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),aa_int_real(ring_1_of_int_real,I4))) )
         => pp(aa_int_bool(P,I4)) ) ) ).

% ceiling_split
tff(fact_5702_ceiling__split,axiom,
    ! [P: fun_int_bool,T: rat] :
      ( pp(aa_int_bool(P,aa_rat_int(archim2889992004027027881ng_rat,T)))
    <=> ! [I4: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,I4)),one_one_rat)),T))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),aa_int_rat(ring_1_of_int_rat,I4))) )
         => pp(aa_int_bool(P,I4)) ) ) ).

% ceiling_split
tff(fact_5703_ceiling__less__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),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_5704_ceiling__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))) ) ).

% ceiling_less_iff
tff(fact_5705_le__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X)) ) ).

% le_ceiling_iff
tff(fact_5706_le__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X)) ) ).

% le_ceiling_iff
tff(fact_5707_divide__le__eq__numeral_I1_J,axiom,
    ! [B3: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)),aa_num_real(numeral_numeral_real,W2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),B3)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W2))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_5708_divide__le__eq__numeral_I1_J,axiom,
    ! [B3: rat,C: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)),aa_num_rat(numeral_numeral_rat,W2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C)),B3)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_5709_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),B3)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,W2)),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_5710_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C)),B3)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_5711_divide__less__eq__numeral_I2_J,axiom,
    ! [B3: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),B3)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5712_divide__less__eq__numeral_I2_J,axiom,
    ! [B3: rat,C: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C)),B3)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5713_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),B3)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5714_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C)),B3)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5715_real__of__int__div2,axiom,
    ! [N: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X))))) ).

% real_of_int_div2
tff(fact_5716_split__zmod,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K)))
    <=> ( ( ( K = zero_zero_int )
         => pp(aa_int_bool(P,N)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
         => ! [I4: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => pp(aa_int_bool(P,J3)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I4: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => pp(aa_int_bool(P,J3)) ) ) ) ) ).

% split_zmod
tff(fact_5717_int__mod__neg__eq,axiom,
    ! [A2: int,B3: int,Q3: int,R2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q3)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),R2))
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B3) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_5718_int__mod__pos__eq,axiom,
    ! [A2: int,B3: int,Q3: int,R2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q3)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B3))
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B3) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_5719_real__of__int__div3,axiom,
    ! [N: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X)))),one_one_real)) ).

% real_of_int_div3
tff(fact_5720_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),one_one_int)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_5721_zmod__minus1,axiom,
    ! [B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),B3) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B3),one_one_int) ) ) ).

% zmod_minus1
tff(fact_5722_zmod__zmult2__eq,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B3),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B3)) ) ) ).

% zmod_zmult2_eq
tff(fact_5723_ceiling__divide__upper,axiom,
    ! [Q3: real,P3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),Q3)))),Q3))) ) ).

% ceiling_divide_upper
tff(fact_5724_ceiling__divide__upper,axiom,
    ! [Q3: rat,P3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,P3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P3),Q3)))),Q3))) ) ).

% ceiling_divide_upper
tff(fact_5725_mult__ceiling__le__Ints,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_set_real_bool(member_real(A2),ring_1_Ints_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A2)),aa_real_int(archim7802044766580827645g_real,B3))))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_5726_mult__ceiling__le__Ints,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_set_real_bool(member_real(A2),ring_1_Ints_real))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A2)),aa_real_int(archim7802044766580827645g_real,B3))))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_5727_mult__ceiling__le__Ints,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_set_real_bool(member_real(A2),ring_1_Ints_real))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B3)))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A2)),aa_real_int(archim7802044766580827645g_real,B3))))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_5728_mult__ceiling__le__Ints,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_set_rat_bool(member_rat(A2),ring_1_Ints_rat))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A2)),aa_rat_int(archim2889992004027027881ng_rat,B3))))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_5729_mult__ceiling__le__Ints,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_set_rat_bool(member_rat(A2),ring_1_Ints_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A2)),aa_rat_int(archim2889992004027027881ng_rat,B3))))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_5730_mult__ceiling__le__Ints,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_set_rat_bool(member_rat(A2),ring_1_Ints_rat))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B3)))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A2)),aa_rat_int(archim2889992004027027881ng_rat,B3))))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_5731_divide__le__eq__numeral_I2_J,axiom,
    ! [B3: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),B3)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5732_divide__le__eq__numeral_I2_J,axiom,
    ! [B3: rat,C: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C)),B3)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5733_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B3),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),B3)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5734_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B3),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C)),B3)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5735_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q3)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R2),aa_num_nat(numeral_numeral_nat,L))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q3)),R2) ) ) ) ).

% divmod_step_eq
tff(fact_5736_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q3: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2))
       => ( unique5024387138958732305ep_int(L,product_Pair_int_int(Q3,R2)) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q3)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_num_int(numeral_numeral_int,L))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2))
       => ( unique5024387138958732305ep_int(L,product_Pair_int_int(Q3,R2)) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q3),R2) ) ) ) ).

% divmod_step_eq
tff(fact_5737_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q3: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2))
       => ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q3,R2)) = produc1086072967326762835nteger(plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Q3),one_one_Code_integer),minus_8373710615458151222nteger(R2,numera6620942414471956472nteger(L))) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2))
       => ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q3,R2)) = produc1086072967326762835nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Q3),R2) ) ) ) ).

% divmod_step_eq
tff(fact_5738_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
         => ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_vebt_insert(T,X)),Y))
           => ( pp(aa_nat_bool(vEBT_vebt_member(T),Y))
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_5739_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
         => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,Y)),X)) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_5740_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,X)),X)) ) ) ).

% valid_insert_both_member_options_add
tff(fact_5741_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary) ) ) ) ).

% insert_simp_mima
tff(fact_5742_valid__pres__insert,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
       => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_insert(T,X)),N)) ) ) ).

% valid_pres_insert
tff(fact_5743_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,vEBT_VEBT_set_vebt(T)),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat)))) ) ).

% inrange
tff(fact_5744_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = M ) ) ).

% mod_less
tff(fact_5745_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_5746_Icc__eq__Icc,axiom,
    ! [L: set_int,H: set_int,L3: set_int,H2: set_int] :
      ( ( set_or370866239135849197et_int(L,H) = set_or370866239135849197et_int(L3,H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(L),H))
          & ~ pp(aa_set_int_bool(ord_less_eq_set_int(L3),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_5747_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L3: rat,H2: rat] :
      ( ( set_or633870826150836451st_rat(L,H) = set_or633870826150836451st_rat(L3,H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),H))
          & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L3),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_5748_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L3: num,H2: num] :
      ( ( set_or7049704709247886629st_num(L,H) = set_or7049704709247886629st_num(L3,H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),H))
          & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L3),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_5749_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L3: nat,H2: nat] :
      ( ( set_or1269000886237332187st_nat(L,H) = set_or1269000886237332187st_nat(L3,H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),H))
          & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L3),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_5750_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L3: int,H2: int] :
      ( ( set_or1266510415728281911st_int(L,H) = set_or1266510415728281911st_int(L3,H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),H))
          & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L3),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_5751_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L3: real,H2: real] :
      ( ( set_or1222579329274155063t_real(L,H) = set_or1222579329274155063t_real(L3,H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),H))
          & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L3),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_5752_atLeastAtMost__iff,axiom,
    ! [I: bool,L: bool,U: bool] :
      ( pp(aa_set_bool_bool(member_bool(I),set_or1454250544011482060t_bool(L,U)))
    <=> ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,L),I))
        & pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_5753_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I),set_or4548717258645045905et_nat(L,U)))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,L),I))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_5754_atLeastAtMost__iff,axiom,
    ! [I: set_int,L: set_int,U: set_int] :
      ( member_set_int(I,set_or370866239135849197et_int(L,U))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(L),I))
        & pp(aa_set_int_bool(ord_less_eq_set_int(I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_5755_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( pp(aa_set_rat_bool(member_rat(I),set_or633870826150836451st_rat(L,U)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),I))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_5756_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( pp(aa_set_num_bool(member_num(I),set_or7049704709247886629st_num(L,U)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),I))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_5757_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),set_or1269000886237332187st_nat(L,U)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),I))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_5758_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( pp(aa_set_int_bool(member_int(I),set_or1266510415728281911st_int(L,U)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),I))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_5759_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( pp(aa_set_real_bool(member_real(I),set_or1222579329274155063t_real(L,U)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),I))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_5760_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,set_or1269000886237332187st_nat(L,U))) ).

% finite_atLeastAtMost
tff(fact_5761_atLeastatMost__empty__iff2,axiom,
    ! [A2: bool,B3: bool] :
      ( ( bot_bot_set_bool = set_or1454250544011482060t_bool(A2,B3) )
    <=> ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,A2),B3)) ) ).

% atLeastatMost_empty_iff2
tff(fact_5762_atLeastatMost__empty__iff2,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( bot_bot_set_set_int = set_or370866239135849197et_int(A2,B3) )
    <=> ~ pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3)) ) ).

% atLeastatMost_empty_iff2
tff(fact_5763_atLeastatMost__empty__iff2,axiom,
    ! [A2: rat,B3: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A2,B3) )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ).

% atLeastatMost_empty_iff2
tff(fact_5764_atLeastatMost__empty__iff2,axiom,
    ! [A2: num,B3: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A2,B3) )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3)) ) ).

% atLeastatMost_empty_iff2
tff(fact_5765_atLeastatMost__empty__iff2,axiom,
    ! [A2: nat,B3: nat] :
      ( ( bot_bot_set_nat = set_or1269000886237332187st_nat(A2,B3) )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3)) ) ).

% atLeastatMost_empty_iff2
tff(fact_5766_atLeastatMost__empty__iff2,axiom,
    ! [A2: int,B3: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(A2,B3) )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ).

% atLeastatMost_empty_iff2
tff(fact_5767_atLeastatMost__empty__iff2,axiom,
    ! [A2: real,B3: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A2,B3) )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ).

% atLeastatMost_empty_iff2
tff(fact_5768_atLeastatMost__empty__iff,axiom,
    ! [A2: bool,B3: bool] :
      ( ( set_or1454250544011482060t_bool(A2,B3) = bot_bot_set_bool )
    <=> ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,A2),B3)) ) ).

% atLeastatMost_empty_iff
tff(fact_5769_atLeastatMost__empty__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( set_or370866239135849197et_int(A2,B3) = bot_bot_set_set_int )
    <=> ~ pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3)) ) ).

% atLeastatMost_empty_iff
tff(fact_5770_atLeastatMost__empty__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( set_or633870826150836451st_rat(A2,B3) = bot_bot_set_rat )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ).

% atLeastatMost_empty_iff
tff(fact_5771_atLeastatMost__empty__iff,axiom,
    ! [A2: num,B3: num] :
      ( ( set_or7049704709247886629st_num(A2,B3) = bot_bot_set_num )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3)) ) ).

% atLeastatMost_empty_iff
tff(fact_5772_atLeastatMost__empty__iff,axiom,
    ! [A2: nat,B3: nat] :
      ( ( set_or1269000886237332187st_nat(A2,B3) = bot_bot_set_nat )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3)) ) ).

% atLeastatMost_empty_iff
tff(fact_5773_atLeastatMost__empty__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( set_or1266510415728281911st_int(A2,B3) = bot_bot_set_int )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ).

% atLeastatMost_empty_iff
tff(fact_5774_atLeastatMost__empty__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( set_or1222579329274155063t_real(A2,B3) = bot_bot_set_real )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3)) ) ).

% atLeastatMost_empty_iff
tff(fact_5775_atLeastatMost__subset__iff,axiom,
    ! [A2: set_int,B3: set_int,C: set_int,D: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(A2,B3),set_or370866239135849197et_int(C,D))
    <=> ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
        | ( pp(aa_set_int_bool(ord_less_eq_set_int(C),A2))
          & pp(aa_set_int_bool(ord_less_eq_set_int(B3),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5776_atLeastatMost__subset__iff,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or633870826150836451st_rat(A2,B3)),set_or633870826150836451st_rat(C,D)))
    <=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5777_atLeastatMost__subset__iff,axiom,
    ! [A2: num,B3: num,C: num,D: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or7049704709247886629st_num(A2,B3)),set_or7049704709247886629st_num(C,D)))
    <=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
        | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A2))
          & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5778_atLeastatMost__subset__iff,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_or1269000886237332187st_nat(A2,B3)),set_or1269000886237332187st_nat(C,D)))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5779_atLeastatMost__subset__iff,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or1266510415728281911st_int(A2,B3)),set_or1266510415728281911st_int(C,D)))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5780_atLeastatMost__subset__iff,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or1222579329274155063t_real(A2,B3)),set_or1222579329274155063t_real(C,D)))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5781_atLeastatMost__empty,axiom,
    ! [B3: bool,A2: bool] :
      ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,B3),A2))
     => ( set_or1454250544011482060t_bool(A2,B3) = bot_bot_set_bool ) ) ).

% atLeastatMost_empty
tff(fact_5782_atLeastatMost__empty,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
     => ( set_or633870826150836451st_rat(A2,B3) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_5783_atLeastatMost__empty,axiom,
    ! [B3: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),A2))
     => ( set_or7049704709247886629st_num(A2,B3) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_5784_atLeastatMost__empty,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
     => ( set_or1269000886237332187st_nat(A2,B3) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_5785_atLeastatMost__empty,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
     => ( set_or1266510415728281911st_int(A2,B3) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_5786_atLeastatMost__empty,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
     => ( set_or1222579329274155063t_real(A2,B3) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_5787_infinite__Icc__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ~ pp(aa_set_rat_bool(finite_finite_rat,set_or633870826150836451st_rat(A2,B3)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3)) ) ).

% infinite_Icc_iff
tff(fact_5788_infinite__Icc__iff,axiom,
    ! [A2: real,B3: real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,set_or1222579329274155063t_real(A2,B3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3)) ) ).

% infinite_Icc_iff
tff(fact_5789_atLeastAtMost__singleton,axiom,
    ! [A2: bool] : set_or1454250544011482060t_bool(A2,A2) = insert_bool(A2,bot_bot_set_bool) ).

% atLeastAtMost_singleton
tff(fact_5790_atLeastAtMost__singleton,axiom,
    ! [A2: nat] : set_or1269000886237332187st_nat(A2,A2) = insert_nat(A2,bot_bot_set_nat) ).

% atLeastAtMost_singleton
tff(fact_5791_atLeastAtMost__singleton,axiom,
    ! [A2: int] : set_or1266510415728281911st_int(A2,A2) = insert_int(A2,bot_bot_set_int) ).

% atLeastAtMost_singleton
tff(fact_5792_atLeastAtMost__singleton,axiom,
    ! [A2: real] : set_or1222579329274155063t_real(A2,A2) = insert_real(A2,bot_bot_set_real) ).

% atLeastAtMost_singleton
tff(fact_5793_atLeastAtMost__singleton__iff,axiom,
    ! [A2: bool,B3: bool,C: bool] :
      ( ( set_or1454250544011482060t_bool(A2,B3) = insert_bool(C,bot_bot_set_bool) )
    <=> ( ( pp(A2)
        <=> pp(B3) )
        & ( pp(B3)
        <=> pp(C) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5794_atLeastAtMost__singleton__iff,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( set_or1269000886237332187st_nat(A2,B3) = insert_nat(C,bot_bot_set_nat) )
    <=> ( ( A2 = B3 )
        & ( B3 = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5795_atLeastAtMost__singleton__iff,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( set_or1266510415728281911st_int(A2,B3) = insert_int(C,bot_bot_set_int) )
    <=> ( ( A2 = B3 )
        & ( B3 = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5796_atLeastAtMost__singleton__iff,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( set_or1222579329274155063t_real(A2,B3) = insert_real(C,bot_bot_set_real) )
    <=> ( ( A2 = B3 )
        & ( B3 = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5797_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_num_real(numeral_numeral_real,W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,W2))) ) ).

% real_of_nat_less_numeral_iff
tff(fact_5798_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W2)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,W2)),N)) ) ).

% numeral_less_real_of_nat_iff
tff(fact_5799_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),aa_nat_real(semiri5074537144036343181t_real,M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),M)) ) ).

% numeral_le_real_of_nat_iff
tff(fact_5800_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_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_5801_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_5802_add__self__mod__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_5803_powr__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_num_real(numeral_numeral_real,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,N)) ) ) ).

% powr_numeral
tff(fact_5804_mod2__gr__0,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_5805_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% half_nonnegative_int_iff
tff(fact_5806_half__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% half_negative_int_iff
tff(fact_5807_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),M)) ).

% mod_less_eq_dividend
tff(fact_5808_infinite__Icc,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ~ pp(aa_set_rat_bool(finite_finite_rat,set_or633870826150836451st_rat(A2,B3))) ) ).

% infinite_Icc
tff(fact_5809_infinite__Icc,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ~ pp(aa_set_real_bool(finite_finite_real,set_or1222579329274155063t_real(A2,B3))) ) ).

% infinite_Icc
tff(fact_5810_atLeastAtMost__singleton_H,axiom,
    ! [A2: bool,B3: bool] :
      ( ( pp(A2)
      <=> pp(B3) )
     => ( set_or1454250544011482060t_bool(A2,B3) = insert_bool(A2,bot_bot_set_bool) ) ) ).

% atLeastAtMost_singleton'
tff(fact_5811_atLeastAtMost__singleton_H,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 = B3 )
     => ( set_or1269000886237332187st_nat(A2,B3) = insert_nat(A2,bot_bot_set_nat) ) ) ).

% atLeastAtMost_singleton'
tff(fact_5812_atLeastAtMost__singleton_H,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 = B3 )
     => ( set_or1266510415728281911st_int(A2,B3) = insert_int(A2,bot_bot_set_int) ) ) ).

% atLeastAtMost_singleton'
tff(fact_5813_atLeastAtMost__singleton_H,axiom,
    ! [A2: real,B3: real] :
      ( ( A2 = B3 )
     => ( set_or1222579329274155063t_real(A2,B3) = insert_real(A2,bot_bot_set_real) ) ) ).

% atLeastAtMost_singleton'
tff(fact_5814_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = zero_zero_nat ) )
      & ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) != N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ) ) ) ).

% mod_Suc
tff(fact_5815_all__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N))
         => pp(aa_nat_bool(P,M3)) )
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_or1269000886237332187st_nat(zero_zero_nat,N)))
         => pp(aa_nat_bool(P,X3)) ) ) ).

% all_nat_less
tff(fact_5816_ex__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N))
          & pp(aa_nat_bool(P,M3)) )
    <=> ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_or1269000886237332187st_nat(zero_zero_nat,N)))
          & pp(aa_nat_bool(P,X3)) ) ) ).

% ex_nat_less
tff(fact_5817_mod__induct,axiom,
    ! [P: fun_nat_bool,N: nat,P3: nat,M: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),P3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),P3))
         => ( ! [N2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),P3))
               => ( pp(aa_nat_bool(P,N2))
                 => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),P3))) ) )
           => pp(aa_nat_bool(P,M)) ) ) ) ) ).

% mod_induct
tff(fact_5818_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N)) ) ).

% mod_less_divisor
tff(fact_5819_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [M4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M4),zero_zero_nat))
     => ( ! [M4: nat,N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,N2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M4),N2)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M4),N2)) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ).

% gcd_nat_induct
tff(fact_5820_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,N))),N)) ).

% mod_Suc_le_divisor
tff(fact_5821_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_5822_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

% mod_geq
tff(fact_5823_mod__if,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = M ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ) ).

% mod_if
tff(fact_5824_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

% le_mod_geq
tff(fact_5825_vebt__insert_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info2,zero_zero_nat,Ts2,S2),X) = vEBT_Node(Info2,zero_zero_nat,Ts2,S2) ).

% vebt_insert.simps(2)
tff(fact_5826_atLeastatMost__psubset__iff,axiom,
    ! [A2: set_int,B3: set_int,C: set_int,D: set_int] :
      ( ord_less_set_set_int(set_or370866239135849197et_int(A2,B3),set_or370866239135849197et_int(C,D))
    <=> ( ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
          | ( pp(aa_set_int_bool(ord_less_eq_set_int(C),A2))
            & pp(aa_set_int_bool(ord_less_eq_set_int(B3),D))
            & ( pp(aa_set_int_bool(ord_less_set_int(C),A2))
              | pp(aa_set_int_bool(ord_less_set_int(B3),D)) ) ) )
        & pp(aa_set_int_bool(ord_less_eq_set_int(C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5827_atLeastatMost__psubset__iff,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( pp(aa_set_rat_bool(ord_less_set_rat(set_or633870826150836451st_rat(A2,B3)),set_or633870826150836451st_rat(C,D)))
    <=> ( ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
          | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A2))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),D))
            & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A2))
              | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),D)) ) ) )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5828_atLeastatMost__psubset__iff,axiom,
    ! [A2: num,B3: num,C: num,D: num] :
      ( pp(aa_set_num_bool(ord_less_set_num(set_or7049704709247886629st_num(A2,B3)),set_or7049704709247886629st_num(C,D)))
    <=> ( ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
          | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A2))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),D))
            & ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A2))
              | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),D)) ) ) )
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5829_atLeastatMost__psubset__iff,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,set_or1269000886237332187st_nat(A2,B3)),set_or1269000886237332187st_nat(C,D)))
    <=> ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),D))
            & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A2))
              | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),D)) ) ) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5830_atLeastatMost__psubset__iff,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(set_or1266510415728281911st_int(A2,B3)),set_or1266510415728281911st_int(C,D)))
    <=> ( ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
          | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),D))
            & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A2))
              | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),D)) ) ) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5831_atLeastatMost__psubset__iff,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(set_or1222579329274155063t_real(A2,B3)),set_or1222579329274155063t_real(C,D)))
    <=> ( ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
          | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),D))
            & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A2))
              | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),D)) ) ) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5832_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N)) ) ).

% mod_le_divisor
tff(fact_5833_div__less__mono,axiom,
    ! [A3: nat,B2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A3),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),N) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),N) = zero_zero_nat )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),N))) ) ) ) ) ).

% div_less_mono
tff(fact_5834_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q3) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ~ ! [S3: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S3)) ) ) ).

% mod_eq_nat1E
tff(fact_5835_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q3) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ~ ! [S3: nat] : N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S3)) ) ) ).

% mod_eq_nat2E
tff(fact_5836_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
       => ? [Q4: nat] : X = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)) ) ) ).

% nat_mod_eq_lemma
tff(fact_5837_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% atLeast0_atMost_Suc
tff(fact_5838_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( set_or1269000886237332187st_nat(M,N) = insert_nat(M,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_5839_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),set_or1269000886237332187st_nat(M,N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_5840_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( insert_nat(M,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N)) = set_or1269000886237332187st_nat(M,N) ) ) ).

% atLeastAtMost_insertL
tff(fact_5841_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,N4),set_or1269000886237332187st_nat(zero_zero_nat,N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_5842_vebt__insert_Osimps_I3_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S2),X) = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) ).

% vebt_insert.simps(3)
tff(fact_5843_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A2: bool,B3: bool] :
      ( ( ( X = zero_zero_nat )
       => ( vEBT_vebt_insert(vEBT_Leaf(A2,B3),X) = vEBT_Leaf(fTrue,B3) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( vEBT_vebt_insert(vEBT_Leaf(A2,B3),X) = vEBT_Leaf(A2,fTrue) ) )
          & ( ( X != one_one_nat )
           => ( vEBT_vebt_insert(vEBT_Leaf(A2,B3),X) = vEBT_Leaf(A2,B3) ) ) ) ) ) ).

% vebt_insert.simps(1)
tff(fact_5844_split__mod,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,M)) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I4)),J3) )
               => pp(aa_nat_bool(P,J3)) ) ) ) ) ) ).

% split_mod
tff(fact_5845_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_5846_two__realpow__ge__one,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N))) ).

% two_realpow_ge_one
tff(fact_5847_nth__rotate1,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => ( aa_nat_int(nth_int(rotate1_int(Xs)),N) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),size_size_list_int(Xs))) ) ) ).

% nth_rotate1
tff(fact_5848_nth__rotate1,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( nth_VEBT_VEBT(rotate1_VEBT_VEBT(Xs),N) = nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),size_s6755466524823107622T_VEBT(Xs))) ) ) ).

% nth_rotate1
tff(fact_5849_nth__rotate1,axiom,
    ! [N: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(nth_bool(rotate1_bool(Xs)),N))
      <=> pp(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),size_size_list_bool(Xs)))) ) ) ).

% nth_rotate1
tff(fact_5850_nth__rotate1,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => ( aa_nat_nat(nth_nat(rotate1_nat(Xs)),N) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),size_size_list_nat(Xs))) ) ) ).

% nth_rotate1
tff(fact_5851_ln__2__less__1,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),one_one_real)) ).

% ln_2_less_1
tff(fact_5852_not__exp__less__eq__0__int,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),zero_zero_int)) ).

% not_exp_less_eq_0_int
tff(fact_5853_verit__le__mono__div,axiom,
    ! [A3: nat,B2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A3),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),N)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),N)),zero_zero_nat)),one_one_nat),zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),N))) ) ) ).

% verit_le_mono_div
tff(fact_5854_exp__half__le2,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% exp_half_le2
tff(fact_5855_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C: real,B3: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),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,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% L2_set_mult_ineq_lemma
tff(fact_5856_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% less_log2_of_power
tff(fact_5857_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% le_log2_of_power
tff(fact_5858_powr__neg__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,N))) ) ) ).

% powr_neg_numeral
tff(fact_5859_pos__zdiv__mult__2,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_5860_neg__zdiv__mult__2,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),one_one_int)),A2) ) ) ).

% neg_zdiv_mult_2
tff(fact_5861_pos__zmod__mult__2,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),A2))) ) ) ).

% pos_zmod_mult_2
tff(fact_5862_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_less
tff(fact_5863_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)))) ) ) ).

% real_exp_bound_lemma
tff(fact_5864_neg__zmod__mult__2,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B3),one_one_int)),A2))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_5865_pos__eucl__rel__int__mult__2,axiom,
    ! [B3: int,A2: int,Q3: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A2,B3),product_Pair_int_int(Q3,R2)))
       => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3)),product_Pair_int_int(Q3,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2))))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_5866_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_le
tff(fact_5867_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(exp_real,X))) ) ).

% exp_lower_Taylor_quadratic
tff(fact_5868_neg__eucl__rel__int__mult__2,axiom,
    ! [B3: int,A2: int,Q3: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),zero_zero_int))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int),B3),product_Pair_int_int(Q3,R2)))
       => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3)),product_Pair_int_int(Q3,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)),one_one_int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_5869_arctan__double,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,X)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_5870_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_5871_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_5872_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_5873_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_5874_VEBT__internal_Oinsert_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_insert(X,Xa2) = Y )
     => ( ! [A: bool,B: bool] :
            ( ( X = vEBT_Leaf(A,B) )
           => ( Y != vEBT_vebt_insert(vEBT_Leaf(A,B),Xa2) ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
             => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)),Xa2))
                   => ( Y = vEBT_Node(Info,Deg2,TreeList,Summary2) ) )
                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)),Xa2))
                   => ( Y = vEBT_vebt_insert(vEBT_Node(Info,Deg2,TreeList,Summary2),Xa2) ) ) ) ) ) ) ).

% VEBT_internal.insert'.elims
tff(fact_5875_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
    ! [Deg: nat,X: nat,Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)),X))
       => ( vEBT_VEBT_insert(vEBT_Node(Info2,Deg,TreeList2,Summary),X) = vEBT_Node(Info2,Deg,TreeList2,Summary) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)),X))
       => ( vEBT_VEBT_insert(vEBT_Node(Info2,Deg,TreeList2,Summary),X) = vEBT_vebt_insert(vEBT_Node(Info2,Deg,TreeList2,Summary),X) ) ) ) ).

% VEBT_internal.insert'.simps(2)
tff(fact_5876_insert__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vEBT_set_vebt(T)),insert_nat(X,bot_bot_set_nat)) = vEBT_set_vebt(vEBT_vebt_insert(T,X)) ) ) ) ).

% insert_correct
tff(fact_5877_insert__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vEBT_VEBT_set_vebt(T)),insert_nat(X,bot_bot_set_nat)) = vEBT_VEBT_set_vebt(vEBT_vebt_insert(T,X)) ) ) ) ).

% insert_corr
tff(fact_5878_abs__sqrt__wlog,axiom,
    ! [P: fun_Co8310303586393106575r_bool,X: code_integer] :
      ( ! [X5: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,X5),aa_nat_Code_integer(power_8256067586552552935nteger(X5),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,abs_abs_Code_integer(X)),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_5879_abs__sqrt__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,X: real] :
      ( ! [X5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,X5),aa_nat_real(aa_real_fun_nat_real(power_power_real,X5),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,aa_real_real(abs_abs_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_5880_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,X: rat] :
      ( ! [X5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,X5),aa_nat_rat(power_power_rat(X5),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,aa_rat_rat(abs_abs_rat,X)),aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_5881_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_bool,X: int] :
      ( ! [X5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,X5),aa_nat_int(power_power_int(X5),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(abs_abs_int,X)),aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_5882_pred__list__to__short,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = none_nat ) ) ) ) ).

% pred_list_to_short
tff(fact_5883_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_5884_UnCI,axiom,
    ! [C: real,B2: set_real,A3: set_real] :
      ( ( ~ pp(aa_set_real_bool(member_real(C),B2))
       => pp(aa_set_real_bool(member_real(C),A3)) )
     => pp(aa_set_real_bool(member_real(C),sup_sup_set_real(A3,B2))) ) ).

% UnCI
tff(fact_5885_UnCI,axiom,
    ! [C: bool,B2: set_bool,A3: set_bool] :
      ( ( ~ pp(aa_set_bool_bool(member_bool(C),B2))
       => pp(aa_set_bool_bool(member_bool(C),A3)) )
     => pp(aa_set_bool_bool(member_bool(C),sup_sup_set_bool(A3,B2))) ) ).

% UnCI
tff(fact_5886_UnCI,axiom,
    ! [C: set_nat,B2: set_set_nat,A3: set_set_nat] :
      ( ( ~ pp(aa_set_set_nat_bool(member_set_nat(C),B2))
       => pp(aa_set_set_nat_bool(member_set_nat(C),A3)) )
     => pp(aa_set_set_nat_bool(member_set_nat(C),sup_sup_set_set_nat(A3,B2))) ) ).

% UnCI
tff(fact_5887_UnCI,axiom,
    ! [C: int,B2: set_int,A3: set_int] :
      ( ( ~ pp(aa_set_int_bool(member_int(C),B2))
       => pp(aa_set_int_bool(member_int(C),A3)) )
     => pp(aa_set_int_bool(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2))) ) ).

% UnCI
tff(fact_5888_UnCI,axiom,
    ! [C: nat,B2: set_nat,A3: set_nat] :
      ( ( ~ pp(aa_set_nat_bool(member_nat(C),B2))
       => pp(aa_set_nat_bool(member_nat(C),A3)) )
     => pp(aa_set_nat_bool(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2))) ) ).

% UnCI
tff(fact_5889_UnCI,axiom,
    ! [C: produc859450856879609959at_nat,B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( ~ member8206827879206165904at_nat(C,B2)
       => member8206827879206165904at_nat(C,A3) )
     => member8206827879206165904at_nat(C,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2)) ) ).

% UnCI
tff(fact_5890_UnCI,axiom,
    ! [C: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ~ member8757157785044589968at_nat(C,B2)
       => member8757157785044589968at_nat(C,A3) )
     => member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2)) ) ).

% UnCI
tff(fact_5891_Un__iff,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),sup_sup_set_real(A3,B2)))
    <=> ( pp(aa_set_real_bool(member_real(C),A3))
        | pp(aa_set_real_bool(member_real(C),B2)) ) ) ).

% Un_iff
tff(fact_5892_Un__iff,axiom,
    ! [C: bool,A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(C),sup_sup_set_bool(A3,B2)))
    <=> ( pp(aa_set_bool_bool(member_bool(C),A3))
        | pp(aa_set_bool_bool(member_bool(C),B2)) ) ) ).

% Un_iff
tff(fact_5893_Un__iff,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),sup_sup_set_set_nat(A3,B2)))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(C),A3))
        | pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ) ).

% Un_iff
tff(fact_5894_Un__iff,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)))
    <=> ( pp(aa_set_int_bool(member_int(C),A3))
        | pp(aa_set_int_bool(member_int(C),B2)) ) ) ).

% Un_iff
tff(fact_5895_Un__iff,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)))
    <=> ( pp(aa_set_nat_bool(member_nat(C),A3))
        | pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).

% Un_iff
tff(fact_5896_Un__iff,axiom,
    ! [C: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( member8206827879206165904at_nat(C,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2))
    <=> ( member8206827879206165904at_nat(C,A3)
        | member8206827879206165904at_nat(C,B2) ) ) ).

% Un_iff
tff(fact_5897_Un__iff,axiom,
    ! [C: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2))
    <=> ( member8757157785044589968at_nat(C,A3)
        | member8757157785044589968at_nat(C,B2) ) ) ).

% Un_iff
tff(fact_5898_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,set_or1266510415728281911st_int(L,U))) ).

% finite_atLeastAtMost_int
tff(fact_5899_high__def,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% high_def
tff(fact_5900_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ).

% high_bound_aux
tff(fact_5901_high__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),X)),N) = Y ) ) ).

% high_inv
tff(fact_5902_Un__empty,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2) = bot_bo5327735625951526323at_nat )
    <=> ( ( A3 = bot_bo5327735625951526323at_nat )
        & ( B2 = bot_bo5327735625951526323at_nat ) ) ) ).

% Un_empty
tff(fact_5903_Un__empty,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2) = bot_bo228742789529271731at_nat )
    <=> ( ( A3 = bot_bo228742789529271731at_nat )
        & ( B2 = bot_bo228742789529271731at_nat ) ) ) ).

% Un_empty
tff(fact_5904_Un__empty,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ( sup_sup_set_real(A3,B2) = bot_bot_set_real )
    <=> ( ( A3 = bot_bot_set_real )
        & ( B2 = bot_bot_set_real ) ) ) ).

% Un_empty
tff(fact_5905_Un__empty,axiom,
    ! [A3: set_bool,B2: set_bool] :
      ( ( sup_sup_set_bool(A3,B2) = bot_bot_set_bool )
    <=> ( ( A3 = bot_bot_set_bool )
        & ( B2 = bot_bot_set_bool ) ) ) ).

% Un_empty
tff(fact_5906_Un__empty,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2) = bot_bot_set_nat )
    <=> ( ( A3 = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% Un_empty
tff(fact_5907_Un__empty,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2) = bot_bot_set_int )
    <=> ( ( A3 = bot_bot_set_int )
        & ( B2 = bot_bot_set_int ) ) ) ).

% Un_empty
tff(fact_5908_finite__Un,axiom,
    ! [F2: set_int,G2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,F2),G2)))
    <=> ( pp(aa_set_int_bool(finite_finite_int,F2))
        & pp(aa_set_int_bool(finite_finite_int,G2)) ) ) ).

% finite_Un
tff(fact_5909_finite__Un,axiom,
    ! [F2: set_complex,G2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,sup_sup_set_complex(F2,G2)))
    <=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F2))
        & pp(aa_set_complex_bool(finite3207457112153483333omplex,G2)) ) ) ).

% finite_Un
tff(fact_5910_finite__Un,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,F2),G2)))
    <=> ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F2))
        & pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,G2)) ) ) ).

% finite_Un
tff(fact_5911_finite__Un,axiom,
    ! [F2: set_Extended_enat,G2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,sup_su4489774667511045786d_enat(F2,G2)))
    <=> ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F2))
        & pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,G2)) ) ) ).

% finite_Un
tff(fact_5912_finite__Un,axiom,
    ! [F2: set_nat,G2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,F2),G2)))
    <=> ( pp(aa_set_nat_bool(finite_finite_nat,F2))
        & pp(aa_set_nat_bool(finite_finite_nat,G2)) ) ) ).

% finite_Un
tff(fact_5913_finite__Un,axiom,
    ! [F2: set_Pr8693737435421807431at_nat,G2: set_Pr8693737435421807431at_nat] :
      ( finite4392333629123659920at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,F2),G2))
    <=> ( finite4392333629123659920at_nat(F2)
        & finite4392333629123659920at_nat(G2) ) ) ).

% finite_Un
tff(fact_5914_finite__Un,axiom,
    ! [F2: set_Pr4329608150637261639at_nat,G2: set_Pr4329608150637261639at_nat] :
      ( finite4343798906461161616at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,F2),G2))
    <=> ( finite4343798906461161616at_nat(F2)
        & finite4343798906461161616at_nat(G2) ) ) ).

% finite_Un
tff(fact_5915_Un__subset__iff,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)),C2))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),C2))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),C2)) ) ) ).

% Un_subset_iff
tff(fact_5916_Un__subset__iff,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2),C2)
    <=> ( ord_le3000389064537975527at_nat(A3,C2)
        & ord_le3000389064537975527at_nat(B2,C2) ) ) ).

% Un_subset_iff
tff(fact_5917_Un__subset__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2),C2)
    <=> ( ord_le1268244103169919719at_nat(A3,C2)
        & ord_le1268244103169919719at_nat(B2,C2) ) ) ).

% Un_subset_iff
tff(fact_5918_Un__subset__iff,axiom,
    ! [A3: set_int,B2: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)),C2))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),C2))
        & pp(aa_set_int_bool(ord_less_eq_set_int(B2),C2)) ) ) ).

% Un_subset_iff
tff(fact_5919_Un__insert__left,axiom,
    ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,insert8211810215607154385at_nat(A2,B2)),C2) = insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C2)) ).

% Un_insert_left
tff(fact_5920_Un__insert__left,axiom,
    ! [A2: real,B2: set_real,C2: set_real] : sup_sup_set_real(insert_real(A2,B2),C2) = insert_real(A2,sup_sup_set_real(B2,C2)) ).

% Un_insert_left
tff(fact_5921_Un__insert__left,axiom,
    ! [A2: bool,B2: set_bool,C2: set_bool] : sup_sup_set_bool(insert_bool(A2,B2),C2) = insert_bool(A2,sup_sup_set_bool(B2,C2)) ).

% Un_insert_left
tff(fact_5922_Un__insert__left,axiom,
    ! [A2: int,B2: set_int,C2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,insert_int(A2,B2)),C2) = insert_int(A2,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B2),C2)) ).

% Un_insert_left
tff(fact_5923_Un__insert__left,axiom,
    ! [A2: nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,insert_nat(A2,B2)),C2) = insert_nat(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) ).

% Un_insert_left
tff(fact_5924_Un__insert__left,axiom,
    ! [A2: produc859450856879609959at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,insert5050368324300391991at_nat(A2,B2)),C2) = insert5050368324300391991at_nat(A2,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B2),C2)) ).

% Un_insert_left
tff(fact_5925_Un__insert__left,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,insert9069300056098147895at_nat(A2,B2)),C2) = insert9069300056098147895at_nat(A2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2)) ).

% Un_insert_left
tff(fact_5926_Un__insert__right,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),insert8211810215607154385at_nat(A2,B2)) = insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B2)) ).

% Un_insert_right
tff(fact_5927_Un__insert__right,axiom,
    ! [A3: set_real,A2: real,B2: set_real] : sup_sup_set_real(A3,insert_real(A2,B2)) = insert_real(A2,sup_sup_set_real(A3,B2)) ).

% Un_insert_right
tff(fact_5928_Un__insert__right,axiom,
    ! [A3: set_bool,A2: bool,B2: set_bool] : sup_sup_set_bool(A3,insert_bool(A2,B2)) = insert_bool(A2,sup_sup_set_bool(A3,B2)) ).

% Un_insert_right
tff(fact_5929_Un__insert__right,axiom,
    ! [A3: set_int,A2: int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),insert_int(A2,B2)) = insert_int(A2,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)) ).

% Un_insert_right
tff(fact_5930_Un__insert__right,axiom,
    ! [A3: set_nat,A2: nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),insert_nat(A2,B2)) = insert_nat(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)) ).

% Un_insert_right
tff(fact_5931_Un__insert__right,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,A2: produc859450856879609959at_nat,B2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),insert5050368324300391991at_nat(A2,B2)) = insert5050368324300391991at_nat(A2,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2)) ).

% Un_insert_right
tff(fact_5932_Un__insert__right,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),insert9069300056098147895at_nat(A2,B2)) = insert9069300056098147895at_nat(A2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2)) ).

% Un_insert_right
tff(fact_5933_Un__Diff__cancel,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,B2),A3)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2) ).

% Un_Diff_cancel
tff(fact_5934_Un__Diff__cancel,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B2),A3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2) ).

% Un_Diff_cancel
tff(fact_5935_Un__Diff__cancel,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2) ).

% Un_Diff_cancel
tff(fact_5936_Un__Diff__cancel2,axiom,
    ! [B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,B2),A3)),A3) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B2),A3) ).

% Un_Diff_cancel2
tff(fact_5937_Un__Diff__cancel2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B2),A3)),A3) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),A3) ).

% Un_Diff_cancel2
tff(fact_5938_Un__Diff__cancel2,axiom,
    ! [B2: set_nat,A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A3)),A3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),A3) ).

% Un_Diff_cancel2
tff(fact_5939_Compl__Diff__eq,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : uminus4384627049435823934at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),B2)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,uminus4384627049435823934at_nat(A3)),B2) ).

% Compl_Diff_eq
tff(fact_5940_Compl__Diff__eq,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),B2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(A3)),B2) ).

% Compl_Diff_eq
tff(fact_5941_Compl__Diff__eq,axiom,
    ! [A3: set_nat,B2: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(A3)),B2) ).

% Compl_Diff_eq
tff(fact_5942_case4_I11_J,axiom,
    ( ( mi != ma )
   => ! [I3: nat] :
        ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m)))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na) = I3 )
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na))) )
          & ! [X4: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),na) = I3 )
                & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),na))) )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,mi),X4))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),ma)) ) ) ) ) ) ).

% case4(11)
tff(fact_5943_UnE,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),sup_sup_set_real(A3,B2)))
     => ( ~ pp(aa_set_real_bool(member_real(C),A3))
       => pp(aa_set_real_bool(member_real(C),B2)) ) ) ).

% UnE
tff(fact_5944_UnE,axiom,
    ! [C: bool,A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(C),sup_sup_set_bool(A3,B2)))
     => ( ~ pp(aa_set_bool_bool(member_bool(C),A3))
       => pp(aa_set_bool_bool(member_bool(C),B2)) ) ) ).

% UnE
tff(fact_5945_UnE,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),sup_sup_set_set_nat(A3,B2)))
     => ( ~ pp(aa_set_set_nat_bool(member_set_nat(C),A3))
       => pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ) ).

% UnE
tff(fact_5946_UnE,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)))
     => ( ~ pp(aa_set_int_bool(member_int(C),A3))
       => pp(aa_set_int_bool(member_int(C),B2)) ) ) ).

% UnE
tff(fact_5947_UnE,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)))
     => ( ~ pp(aa_set_nat_bool(member_nat(C),A3))
       => pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).

% UnE
tff(fact_5948_UnE,axiom,
    ! [C: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( member8206827879206165904at_nat(C,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2))
     => ( ~ member8206827879206165904at_nat(C,A3)
       => member8206827879206165904at_nat(C,B2) ) ) ).

% UnE
tff(fact_5949_UnE,axiom,
    ! [C: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2))
     => ( ~ member8757157785044589968at_nat(C,A3)
       => member8757157785044589968at_nat(C,B2) ) ) ).

% UnE
tff(fact_5950_UnI1,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),A3))
     => pp(aa_set_real_bool(member_real(C),sup_sup_set_real(A3,B2))) ) ).

% UnI1
tff(fact_5951_UnI1,axiom,
    ! [C: bool,A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(C),A3))
     => pp(aa_set_bool_bool(member_bool(C),sup_sup_set_bool(A3,B2))) ) ).

% UnI1
tff(fact_5952_UnI1,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),A3))
     => pp(aa_set_set_nat_bool(member_set_nat(C),sup_sup_set_set_nat(A3,B2))) ) ).

% UnI1
tff(fact_5953_UnI1,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),A3))
     => pp(aa_set_int_bool(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2))) ) ).

% UnI1
tff(fact_5954_UnI1,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),A3))
     => pp(aa_set_nat_bool(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2))) ) ).

% UnI1
tff(fact_5955_UnI1,axiom,
    ! [C: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( member8206827879206165904at_nat(C,A3)
     => member8206827879206165904at_nat(C,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2)) ) ).

% UnI1
tff(fact_5956_UnI1,axiom,
    ! [C: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,A3)
     => member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2)) ) ).

% UnI1
tff(fact_5957_UnI2,axiom,
    ! [C: real,B2: set_real,A3: set_real] :
      ( pp(aa_set_real_bool(member_real(C),B2))
     => pp(aa_set_real_bool(member_real(C),sup_sup_set_real(A3,B2))) ) ).

% UnI2
tff(fact_5958_UnI2,axiom,
    ! [C: bool,B2: set_bool,A3: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(C),B2))
     => pp(aa_set_bool_bool(member_bool(C),sup_sup_set_bool(A3,B2))) ) ).

% UnI2
tff(fact_5959_UnI2,axiom,
    ! [C: set_nat,B2: set_set_nat,A3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),B2))
     => pp(aa_set_set_nat_bool(member_set_nat(C),sup_sup_set_set_nat(A3,B2))) ) ).

% UnI2
tff(fact_5960_UnI2,axiom,
    ! [C: int,B2: set_int,A3: set_int] :
      ( pp(aa_set_int_bool(member_int(C),B2))
     => pp(aa_set_int_bool(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2))) ) ).

% UnI2
tff(fact_5961_UnI2,axiom,
    ! [C: nat,B2: set_nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),B2))
     => pp(aa_set_nat_bool(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2))) ) ).

% UnI2
tff(fact_5962_UnI2,axiom,
    ! [C: produc859450856879609959at_nat,B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( member8206827879206165904at_nat(C,B2)
     => member8206827879206165904at_nat(C,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2)) ) ).

% UnI2
tff(fact_5963_UnI2,axiom,
    ! [C: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,B2)
     => member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2)) ) ).

% UnI2
tff(fact_5964_bex__Un,axiom,
    ! [A3: set_nat,B2: set_nat,P: fun_nat_bool] :
      ( ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)))
          & pp(aa_nat_bool(P,X3)) )
    <=> ( ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A3))
            & pp(aa_nat_bool(P,X3)) )
        | ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),B2))
            & pp(aa_nat_bool(P,X3)) ) ) ) ).

% bex_Un
tff(fact_5965_bex__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,P: fun_Pr446604468264085555t_bool] :
      ( ? [X3: produc859450856879609959at_nat] :
          ( member8206827879206165904at_nat(X3,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2))
          & pp(aa_Pro6098352494050694700t_bool(P,X3)) )
    <=> ( ? [X3: produc859450856879609959at_nat] :
            ( member8206827879206165904at_nat(X3,A3)
            & pp(aa_Pro6098352494050694700t_bool(P,X3)) )
        | ? [X3: produc859450856879609959at_nat] :
            ( member8206827879206165904at_nat(X3,B2)
            & pp(aa_Pro6098352494050694700t_bool(P,X3)) ) ) ) ).

% bex_Un
tff(fact_5966_bex__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,P: fun_Pr1607443439640623155t_bool] :
      ( ? [X3: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X3,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2))
          & pp(aa_Pro1809015098595223084t_bool(P,X3)) )
    <=> ( ? [X3: produc3843707927480180839at_nat] :
            ( member8757157785044589968at_nat(X3,A3)
            & pp(aa_Pro1809015098595223084t_bool(P,X3)) )
        | ? [X3: produc3843707927480180839at_nat] :
            ( member8757157785044589968at_nat(X3,B2)
            & pp(aa_Pro1809015098595223084t_bool(P,X3)) ) ) ) ).

% bex_Un
tff(fact_5967_ball__Un,axiom,
    ! [A3: set_nat,B2: set_nat,P: fun_nat_bool] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)))
         => pp(aa_nat_bool(P,X3)) )
    <=> ( ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A3))
           => pp(aa_nat_bool(P,X3)) )
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),B2))
           => pp(aa_nat_bool(P,X3)) ) ) ) ).

% ball_Un
tff(fact_5968_ball__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,P: fun_Pr446604468264085555t_bool] :
      ( ! [X3: produc859450856879609959at_nat] :
          ( member8206827879206165904at_nat(X3,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2))
         => pp(aa_Pro6098352494050694700t_bool(P,X3)) )
    <=> ( ! [X3: produc859450856879609959at_nat] :
            ( member8206827879206165904at_nat(X3,A3)
           => pp(aa_Pro6098352494050694700t_bool(P,X3)) )
        & ! [X3: produc859450856879609959at_nat] :
            ( member8206827879206165904at_nat(X3,B2)
           => pp(aa_Pro6098352494050694700t_bool(P,X3)) ) ) ) ).

% ball_Un
tff(fact_5969_ball__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,P: fun_Pr1607443439640623155t_bool] :
      ( ! [X3: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X3,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2))
         => pp(aa_Pro1809015098595223084t_bool(P,X3)) )
    <=> ( ! [X3: produc3843707927480180839at_nat] :
            ( member8757157785044589968at_nat(X3,A3)
           => pp(aa_Pro1809015098595223084t_bool(P,X3)) )
        & ! [X3: produc3843707927480180839at_nat] :
            ( member8757157785044589968at_nat(X3,B2)
           => pp(aa_Pro1809015098595223084t_bool(P,X3)) ) ) ) ).

% ball_Un
tff(fact_5970_Un__assoc,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) ).

% Un_assoc
tff(fact_5971_Un__assoc,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2)),C2) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B2),C2)) ).

% Un_assoc
tff(fact_5972_Un__assoc,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2)) ).

% Un_assoc
tff(fact_5973_Un__absorb,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),A3) = A3 ).

% Un_absorb
tff(fact_5974_Un__absorb,axiom,
    ! [A3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),A3) = A3 ).

% Un_absorb
tff(fact_5975_Un__absorb,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),A3) = A3 ).

% Un_absorb
tff(fact_5976_Un__commute,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),A3) ).

% Un_commute
tff(fact_5977_Un__commute,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B2),A3) ).

% Un_commute
tff(fact_5978_Un__commute,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),A3) ).

% Un_commute
tff(fact_5979_Un__left__absorb,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2) ).

% Un_left_absorb
tff(fact_5980_Un__left__absorb,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2) ).

% Un_left_absorb
tff(fact_5981_Un__left__absorb,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2) ).

% Un_left_absorb
tff(fact_5982_Un__left__commute,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),C2)) ).

% Un_left_commute
tff(fact_5983_Un__left__commute,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B2),C2)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B2),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),C2)) ).

% Un_left_commute
tff(fact_5984_Un__left__commute,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),C2)) ).

% Un_left_commute
tff(fact_5985_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),bot_bo5327735625951526323at_nat) = X ).

% boolean_algebra.disj_zero_right
tff(fact_5986_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),bot_bo228742789529271731at_nat) = X ).

% boolean_algebra.disj_zero_right
tff(fact_5987_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_real] : sup_sup_set_real(X,bot_bot_set_real) = X ).

% boolean_algebra.disj_zero_right
tff(fact_5988_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_bool] : sup_sup_set_bool(X,bot_bot_set_bool) = X ).

% boolean_algebra.disj_zero_right
tff(fact_5989_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),bot_bot_set_nat) = X ).

% boolean_algebra.disj_zero_right
tff(fact_5990_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),bot_bot_set_int) = X ).

% boolean_algebra.disj_zero_right
tff(fact_5991_Un__empty__right,axiom,
    ! [A3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),bot_bo5327735625951526323at_nat) = A3 ).

% Un_empty_right
tff(fact_5992_Un__empty__right,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),bot_bo228742789529271731at_nat) = A3 ).

% Un_empty_right
tff(fact_5993_Un__empty__right,axiom,
    ! [A3: set_real] : sup_sup_set_real(A3,bot_bot_set_real) = A3 ).

% Un_empty_right
tff(fact_5994_Un__empty__right,axiom,
    ! [A3: set_bool] : sup_sup_set_bool(A3,bot_bot_set_bool) = A3 ).

% Un_empty_right
tff(fact_5995_Un__empty__right,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),bot_bot_set_nat) = A3 ).

% Un_empty_right
tff(fact_5996_Un__empty__right,axiom,
    ! [A3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),bot_bot_set_int) = A3 ).

% Un_empty_right
tff(fact_5997_Un__empty__left,axiom,
    ! [B2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,bot_bo5327735625951526323at_nat),B2) = B2 ).

% Un_empty_left
tff(fact_5998_Un__empty__left,axiom,
    ! [B2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),B2) = B2 ).

% Un_empty_left
tff(fact_5999_Un__empty__left,axiom,
    ! [B2: set_real] : sup_sup_set_real(bot_bot_set_real,B2) = B2 ).

% Un_empty_left
tff(fact_6000_Un__empty__left,axiom,
    ! [B2: set_bool] : sup_sup_set_bool(bot_bot_set_bool,B2) = B2 ).

% Un_empty_left
tff(fact_6001_Un__empty__left,axiom,
    ! [B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),B2) = B2 ).

% Un_empty_left
tff(fact_6002_Un__empty__left,axiom,
    ! [B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),B2) = B2 ).

% Un_empty_left
tff(fact_6003_infinite__Un,axiom,
    ! [S: set_int,T2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,S),T2)))
    <=> ( ~ pp(aa_set_int_bool(finite_finite_int,S))
        | ~ pp(aa_set_int_bool(finite_finite_int,T2)) ) ) ).

% infinite_Un
tff(fact_6004_infinite__Un,axiom,
    ! [S: set_complex,T2: set_complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,sup_sup_set_complex(S,T2)))
    <=> ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
        | ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,T2)) ) ) ).

% infinite_Un
tff(fact_6005_infinite__Un,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T2)))
    <=> ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S))
        | ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,T2)) ) ) ).

% infinite_Un
tff(fact_6006_infinite__Un,axiom,
    ! [S: set_Extended_enat,T2: set_Extended_enat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,sup_su4489774667511045786d_enat(S,T2)))
    <=> ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
        | ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T2)) ) ) ).

% infinite_Un
tff(fact_6007_infinite__Un,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T2)))
    <=> ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
        | ~ pp(aa_set_nat_bool(finite_finite_nat,T2)) ) ) ).

% infinite_Un
tff(fact_6008_infinite__Un,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
      ( ~ finite4392333629123659920at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,S),T2))
    <=> ( ~ finite4392333629123659920at_nat(S)
        | ~ finite4392333629123659920at_nat(T2) ) ) ).

% infinite_Un
tff(fact_6009_infinite__Un,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
      ( ~ finite4343798906461161616at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T2))
    <=> ( ~ finite4343798906461161616at_nat(S)
        | ~ finite4343798906461161616at_nat(T2) ) ) ).

% infinite_Un
tff(fact_6010_Un__infinite,axiom,
    ! [S: set_int,T2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S))
     => ~ pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,S),T2))) ) ).

% Un_infinite
tff(fact_6011_Un__infinite,axiom,
    ! [S: set_complex,T2: set_complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,sup_sup_set_complex(S,T2))) ) ).

% Un_infinite
tff(fact_6012_Un__infinite,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S))
     => ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T2))) ) ).

% Un_infinite
tff(fact_6013_Un__infinite,axiom,
    ! [S: set_Extended_enat,T2: set_Extended_enat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,sup_su4489774667511045786d_enat(S,T2))) ) ).

% Un_infinite
tff(fact_6014_Un__infinite,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
     => ~ pp(aa_set_nat_bool(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T2))) ) ).

% Un_infinite
tff(fact_6015_Un__infinite,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
      ( ~ finite4392333629123659920at_nat(S)
     => ~ finite4392333629123659920at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,S),T2)) ) ).

% Un_infinite
tff(fact_6016_Un__infinite,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
      ( ~ finite4343798906461161616at_nat(S)
     => ~ finite4343798906461161616at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T2)) ) ).

% Un_infinite
tff(fact_6017_finite__UnI,axiom,
    ! [F2: set_int,G2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,F2))
     => ( pp(aa_set_int_bool(finite_finite_int,G2))
       => pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,F2),G2))) ) ) ).

% finite_UnI
tff(fact_6018_finite__UnI,axiom,
    ! [F2: set_complex,G2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,G2))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,sup_sup_set_complex(F2,G2))) ) ) ).

% finite_UnI
tff(fact_6019_finite__UnI,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F2))
     => ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,G2))
       => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,F2),G2))) ) ) ).

% finite_UnI
tff(fact_6020_finite__UnI,axiom,
    ! [F2: set_Extended_enat,G2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,G2))
       => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,sup_su4489774667511045786d_enat(F2,G2))) ) ) ).

% finite_UnI
tff(fact_6021_finite__UnI,axiom,
    ! [F2: set_nat,G2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,G2))
       => pp(aa_set_nat_bool(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,F2),G2))) ) ) ).

% finite_UnI
tff(fact_6022_finite__UnI,axiom,
    ! [F2: set_Pr8693737435421807431at_nat,G2: set_Pr8693737435421807431at_nat] :
      ( finite4392333629123659920at_nat(F2)
     => ( finite4392333629123659920at_nat(G2)
       => finite4392333629123659920at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,F2),G2)) ) ) ).

% finite_UnI
tff(fact_6023_finite__UnI,axiom,
    ! [F2: set_Pr4329608150637261639at_nat,G2: set_Pr4329608150637261639at_nat] :
      ( finite4343798906461161616at_nat(F2)
     => ( finite4343798906461161616at_nat(G2)
       => finite4343798906461161616at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,F2),G2)) ) ) ).

% finite_UnI
tff(fact_6024_subset__Un__eq,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2) = B2 ) ) ).

% subset_Un_eq
tff(fact_6025_subset__Un__eq,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A3,B2)
    <=> ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2) = B2 ) ) ).

% subset_Un_eq
tff(fact_6026_subset__Un__eq,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A3,B2)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2) = B2 ) ) ).

% subset_Un_eq
tff(fact_6027_subset__Un__eq,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2) = B2 ) ) ).

% subset_Un_eq
tff(fact_6028_subset__UnE,axiom,
    ! [C2: set_nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)))
     => ~ ! [A8: set_nat] :
            ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A8),A3))
           => ! [B10: set_nat] :
                ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B10),B2))
               => ( C2 != aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A8),B10) ) ) ) ) ).

% subset_UnE
tff(fact_6029_subset__UnE,axiom,
    ! [C2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(C2,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2))
     => ~ ! [A8: set_Pr8693737435421807431at_nat] :
            ( ord_le3000389064537975527at_nat(A8,A3)
           => ! [B10: set_Pr8693737435421807431at_nat] :
                ( ord_le3000389064537975527at_nat(B10,B2)
               => ( C2 != aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A8),B10) ) ) ) ) ).

% subset_UnE
tff(fact_6030_subset__UnE,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(C2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2))
     => ~ ! [A8: set_Pr4329608150637261639at_nat] :
            ( ord_le1268244103169919719at_nat(A8,A3)
           => ! [B10: set_Pr4329608150637261639at_nat] :
                ( ord_le1268244103169919719at_nat(B10,B2)
               => ( C2 != aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A8),B10) ) ) ) ) ).

% subset_UnE
tff(fact_6031_subset__UnE,axiom,
    ! [C2: set_int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)))
     => ~ ! [A8: set_int] :
            ( pp(aa_set_int_bool(ord_less_eq_set_int(A8),A3))
           => ! [B10: set_int] :
                ( pp(aa_set_int_bool(ord_less_eq_set_int(B10),B2))
               => ( C2 != aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A8),B10) ) ) ) ) ).

% subset_UnE
tff(fact_6032_Un__absorb2,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2) = A3 ) ) ).

% Un_absorb2
tff(fact_6033_Un__absorb2,axiom,
    ! [B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(B2,A3)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2) = A3 ) ) ).

% Un_absorb2
tff(fact_6034_Un__absorb2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B2,A3)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2) = A3 ) ) ).

% Un_absorb2
tff(fact_6035_Un__absorb2,axiom,
    ! [B2: set_int,A3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A3))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2) = A3 ) ) ).

% Un_absorb2
tff(fact_6036_Un__absorb1,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2) = B2 ) ) ).

% Un_absorb1
tff(fact_6037_Un__absorb1,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A3,B2)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2) = B2 ) ) ).

% Un_absorb1
tff(fact_6038_Un__absorb1,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A3,B2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2) = B2 ) ) ).

% Un_absorb1
tff(fact_6039_Un__absorb1,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2) = B2 ) ) ).

% Un_absorb1
tff(fact_6040_Un__upper2,axiom,
    ! [B2: set_nat,A3: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2))) ).

% Un_upper2
tff(fact_6041_Un__upper2,axiom,
    ! [B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(B2,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2)) ).

% Un_upper2
tff(fact_6042_Un__upper2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(B2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2)) ).

% Un_upper2
tff(fact_6043_Un__upper2,axiom,
    ! [B2: set_int,A3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(B2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2))) ).

% Un_upper2
tff(fact_6044_Un__upper1,axiom,
    ! [A3: set_nat,B2: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2))) ).

% Un_upper1
tff(fact_6045_Un__upper1,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(A3,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2)) ).

% Un_upper1
tff(fact_6046_Un__upper1,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(A3,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2)) ).

% Un_upper1
tff(fact_6047_Un__upper1,axiom,
    ! [A3: set_int,B2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A3),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2))) ).

% Un_upper1
tff(fact_6048_Un__least,axiom,
    ! [A3: set_nat,C2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),C2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),C2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)),C2)) ) ) ).

% Un_least
tff(fact_6049_Un__least,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A3,C2)
     => ( ord_le3000389064537975527at_nat(B2,C2)
       => ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2),C2) ) ) ).

% Un_least
tff(fact_6050_Un__least,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A3,C2)
     => ( ord_le1268244103169919719at_nat(B2,C2)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2),C2) ) ) ).

% Un_least
tff(fact_6051_Un__least,axiom,
    ! [A3: set_int,C2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),C2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)),C2)) ) ) ).

% Un_least
tff(fact_6052_Un__mono,axiom,
    ! [A3: set_nat,C2: set_nat,B2: set_nat,D4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),C2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),D4))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),D4))) ) ) ).

% Un_mono
tff(fact_6053_Un__mono,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,D4: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A3,C2)
     => ( ord_le3000389064537975527at_nat(B2,D4)
       => ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,C2),D4)) ) ) ).

% Un_mono
tff(fact_6054_Un__mono,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,D4: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A3,C2)
     => ( ord_le1268244103169919719at_nat(B2,D4)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C2),D4)) ) ) ).

% Un_mono
tff(fact_6055_Un__mono,axiom,
    ! [A3: set_int,C2: set_int,B2: set_int,D4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),C2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),D4))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,C2),D4))) ) ) ).

% Un_mono
tff(fact_6056_Un__Diff,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2)),C2) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),C2)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,B2),C2)) ).

% Un_Diff
tff(fact_6057_Un__Diff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),C2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B2),C2)) ).

% Un_Diff
tff(fact_6058_Un__Diff,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),C2)) ).

% Un_Diff
tff(fact_6059_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: rat,M: rat,U: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),M))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,M),U))
       => ( sup_sup_set_rat(set_or633870826150836451st_rat(L,M),set_or633870826150836451st_rat(M,U)) = set_or633870826150836451st_rat(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_6060_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: num,M: num,U: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),M))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),U))
       => ( sup_sup_set_num(set_or7049704709247886629st_num(L,M),set_or7049704709247886629st_num(M,U)) = set_or7049704709247886629st_num(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_6061_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),U))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,set_or1269000886237332187st_nat(L,M)),set_or1269000886237332187st_nat(M,U)) = set_or1269000886237332187st_nat(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_6062_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: int,M: int,U: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),U))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,set_or1266510415728281911st_int(L,M)),set_or1266510415728281911st_int(M,U)) = set_or1266510415728281911st_int(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_6063_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: real,M: real,U: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M),U))
       => ( sup_sup_set_real(set_or1222579329274155063t_real(L,M),set_or1222579329274155063t_real(M,U)) = set_or1222579329274155063t_real(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_6064_insert__is__Un,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A2,A3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat)),A3) ).

% insert_is_Un
tff(fact_6065_insert__is__Un,axiom,
    ! [A2: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat] : insert5050368324300391991at_nat(A2,A3) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,insert5050368324300391991at_nat(A2,bot_bo5327735625951526323at_nat)),A3) ).

% insert_is_Un
tff(fact_6066_insert__is__Un,axiom,
    ! [A2: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A2,A3) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,insert9069300056098147895at_nat(A2,bot_bo228742789529271731at_nat)),A3) ).

% insert_is_Un
tff(fact_6067_insert__is__Un,axiom,
    ! [A2: real,A3: set_real] : insert_real(A2,A3) = sup_sup_set_real(insert_real(A2,bot_bot_set_real),A3) ).

% insert_is_Un
tff(fact_6068_insert__is__Un,axiom,
    ! [A2: bool,A3: set_bool] : insert_bool(A2,A3) = sup_sup_set_bool(insert_bool(A2,bot_bot_set_bool),A3) ).

% insert_is_Un
tff(fact_6069_insert__is__Un,axiom,
    ! [A2: nat,A3: set_nat] : insert_nat(A2,A3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,insert_nat(A2,bot_bot_set_nat)),A3) ).

% insert_is_Un
tff(fact_6070_insert__is__Un,axiom,
    ! [A2: int,A3: set_int] : insert_int(A2,A3) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,insert_int(A2,bot_bot_set_int)),A3) ).

% insert_is_Un
tff(fact_6071_Un__singleton__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B2) = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
    <=> ( ( ( A3 = bot_bo2099793752762293965at_nat )
          & ( B2 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) )
        | ( ( A3 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
          & ( B2 = bot_bo2099793752762293965at_nat ) )
        | ( ( A3 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
          & ( B2 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_6072_Un__singleton__iff,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat] :
      ( ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2) = insert5050368324300391991at_nat(X,bot_bo5327735625951526323at_nat) )
    <=> ( ( ( A3 = bot_bo5327735625951526323at_nat )
          & ( B2 = insert5050368324300391991at_nat(X,bot_bo5327735625951526323at_nat) ) )
        | ( ( A3 = insert5050368324300391991at_nat(X,bot_bo5327735625951526323at_nat) )
          & ( B2 = bot_bo5327735625951526323at_nat ) )
        | ( ( A3 = insert5050368324300391991at_nat(X,bot_bo5327735625951526323at_nat) )
          & ( B2 = insert5050368324300391991at_nat(X,bot_bo5327735625951526323at_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_6073_Un__singleton__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2) = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
    <=> ( ( ( A3 = bot_bo228742789529271731at_nat )
          & ( B2 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) )
        | ( ( A3 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
          & ( B2 = bot_bo228742789529271731at_nat ) )
        | ( ( A3 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
          & ( B2 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_6074_Un__singleton__iff,axiom,
    ! [A3: set_real,B2: set_real,X: real] :
      ( ( sup_sup_set_real(A3,B2) = insert_real(X,bot_bot_set_real) )
    <=> ( ( ( A3 = bot_bot_set_real )
          & ( B2 = insert_real(X,bot_bot_set_real) ) )
        | ( ( A3 = insert_real(X,bot_bot_set_real) )
          & ( B2 = bot_bot_set_real ) )
        | ( ( A3 = insert_real(X,bot_bot_set_real) )
          & ( B2 = insert_real(X,bot_bot_set_real) ) ) ) ) ).

% Un_singleton_iff
tff(fact_6075_Un__singleton__iff,axiom,
    ! [A3: set_bool,B2: set_bool,X: bool] :
      ( ( sup_sup_set_bool(A3,B2) = insert_bool(X,bot_bot_set_bool) )
    <=> ( ( ( A3 = bot_bot_set_bool )
          & ( B2 = insert_bool(X,bot_bot_set_bool) ) )
        | ( ( A3 = insert_bool(X,bot_bot_set_bool) )
          & ( B2 = bot_bot_set_bool ) )
        | ( ( A3 = insert_bool(X,bot_bot_set_bool) )
          & ( B2 = insert_bool(X,bot_bot_set_bool) ) ) ) ) ).

% Un_singleton_iff
tff(fact_6076_Un__singleton__iff,axiom,
    ! [A3: set_nat,B2: set_nat,X: nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2) = insert_nat(X,bot_bot_set_nat) )
    <=> ( ( ( A3 = bot_bot_set_nat )
          & ( B2 = insert_nat(X,bot_bot_set_nat) ) )
        | ( ( A3 = insert_nat(X,bot_bot_set_nat) )
          & ( B2 = bot_bot_set_nat ) )
        | ( ( A3 = insert_nat(X,bot_bot_set_nat) )
          & ( B2 = insert_nat(X,bot_bot_set_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_6077_Un__singleton__iff,axiom,
    ! [A3: set_int,B2: set_int,X: int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2) = insert_int(X,bot_bot_set_int) )
    <=> ( ( ( A3 = bot_bot_set_int )
          & ( B2 = insert_int(X,bot_bot_set_int) ) )
        | ( ( A3 = insert_int(X,bot_bot_set_int) )
          & ( B2 = bot_bot_set_int ) )
        | ( ( A3 = insert_int(X,bot_bot_set_int) )
          & ( B2 = insert_int(X,bot_bot_set_int) ) ) ) ) ).

% Un_singleton_iff
tff(fact_6078_singleton__Un__iff,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B2) )
    <=> ( ( ( A3 = bot_bo2099793752762293965at_nat )
          & ( B2 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) )
        | ( ( A3 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
          & ( B2 = bot_bo2099793752762293965at_nat ) )
        | ( ( A3 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
          & ( B2 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_6079_singleton__Un__iff,axiom,
    ! [X: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( insert5050368324300391991at_nat(X,bot_bo5327735625951526323at_nat) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2) )
    <=> ( ( ( A3 = bot_bo5327735625951526323at_nat )
          & ( B2 = insert5050368324300391991at_nat(X,bot_bo5327735625951526323at_nat) ) )
        | ( ( A3 = insert5050368324300391991at_nat(X,bot_bo5327735625951526323at_nat) )
          & ( B2 = bot_bo5327735625951526323at_nat ) )
        | ( ( A3 = insert5050368324300391991at_nat(X,bot_bo5327735625951526323at_nat) )
          & ( B2 = insert5050368324300391991at_nat(X,bot_bo5327735625951526323at_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_6080_singleton__Un__iff,axiom,
    ! [X: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2) )
    <=> ( ( ( A3 = bot_bo228742789529271731at_nat )
          & ( B2 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) )
        | ( ( A3 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
          & ( B2 = bot_bo228742789529271731at_nat ) )
        | ( ( A3 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
          & ( B2 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_6081_singleton__Un__iff,axiom,
    ! [X: real,A3: set_real,B2: set_real] :
      ( ( insert_real(X,bot_bot_set_real) = sup_sup_set_real(A3,B2) )
    <=> ( ( ( A3 = bot_bot_set_real )
          & ( B2 = insert_real(X,bot_bot_set_real) ) )
        | ( ( A3 = insert_real(X,bot_bot_set_real) )
          & ( B2 = bot_bot_set_real ) )
        | ( ( A3 = insert_real(X,bot_bot_set_real) )
          & ( B2 = insert_real(X,bot_bot_set_real) ) ) ) ) ).

% singleton_Un_iff
tff(fact_6082_singleton__Un__iff,axiom,
    ! [X: bool,A3: set_bool,B2: set_bool] :
      ( ( insert_bool(X,bot_bot_set_bool) = sup_sup_set_bool(A3,B2) )
    <=> ( ( ( A3 = bot_bot_set_bool )
          & ( B2 = insert_bool(X,bot_bot_set_bool) ) )
        | ( ( A3 = insert_bool(X,bot_bot_set_bool) )
          & ( B2 = bot_bot_set_bool ) )
        | ( ( A3 = insert_bool(X,bot_bot_set_bool) )
          & ( B2 = insert_bool(X,bot_bot_set_bool) ) ) ) ) ).

% singleton_Un_iff
tff(fact_6083_singleton__Un__iff,axiom,
    ! [X: nat,A3: set_nat,B2: set_nat] :
      ( ( insert_nat(X,bot_bot_set_nat) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2) )
    <=> ( ( ( A3 = bot_bot_set_nat )
          & ( B2 = insert_nat(X,bot_bot_set_nat) ) )
        | ( ( A3 = insert_nat(X,bot_bot_set_nat) )
          & ( B2 = bot_bot_set_nat ) )
        | ( ( A3 = insert_nat(X,bot_bot_set_nat) )
          & ( B2 = insert_nat(X,bot_bot_set_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_6084_singleton__Un__iff,axiom,
    ! [X: int,A3: set_int,B2: set_int] :
      ( ( insert_int(X,bot_bot_set_int) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2) )
    <=> ( ( ( A3 = bot_bot_set_int )
          & ( B2 = insert_int(X,bot_bot_set_int) ) )
        | ( ( A3 = insert_int(X,bot_bot_set_int) )
          & ( B2 = bot_bot_set_int ) )
        | ( ( A3 = insert_int(X,bot_bot_set_int) )
          & ( B2 = insert_int(X,bot_bot_set_int) ) ) ) ) ).

% singleton_Un_iff
tff(fact_6085_Diff__subset__conv,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),B2),C2)
    <=> ord_le3000389064537975527at_nat(A3,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B2),C2)) ) ).

% Diff_subset_conv
tff(fact_6086_Diff__subset__conv,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),B2),C2)
    <=> ord_le1268244103169919719at_nat(A3,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2)) ) ).

% Diff_subset_conv
tff(fact_6087_Diff__subset__conv,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)),C2))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2))) ) ).

% Diff_subset_conv
tff(fact_6088_Diff__subset__conv,axiom,
    ! [A3: set_int,B2: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2)),C2))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(A3),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B2),C2))) ) ).

% Diff_subset_conv
tff(fact_6089_Diff__partition,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A3,B2)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,B2),A3)) = B2 ) ) ).

% Diff_partition
tff(fact_6090_Diff__partition,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A3,B2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B2),A3)) = B2 ) ) ).

% Diff_partition
tff(fact_6091_Diff__partition,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A3)) = B2 ) ) ).

% Diff_partition
tff(fact_6092_Diff__partition,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B2),A3)) = B2 ) ) ).

% Diff_partition
tff(fact_6093_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)))
     => ( set_or1266510415728281911st_int(M,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N),set_or1266510415728281911st_int(M,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_6094_simp__from__to,axiom,
    ! [J: int,I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = bot_bot_set_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = insert_int(I,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ) ).

% simp_from_to
tff(fact_6095_card__Un__le,axiom,
    ! [A3: set_complex,B2: set_complex] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(sup_sup_set_complex(A3,B2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_complex(A3)),finite_card_complex(B2)))) ).

% card_Un_le
tff(fact_6096_card__Un__le,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(sup_sup_set_list_nat(A3,B2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(A3)),finite_card_list_nat(B2)))) ).

% card_Un_le
tff(fact_6097_card__Un__le,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(sup_sup_set_set_nat(A3,B2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_set_nat(A3)),finite_card_set_nat(B2)))) ).

% card_Un_le
tff(fact_6098_card__Un__le,axiom,
    ! [A3: set_int,B2: set_int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_int(A3)),finite_card_int(B2)))) ).

% card_Un_le
tff(fact_6099_card__Un__le,axiom,
    ! [A3: set_nat,B2: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,B2)))) ).

% card_Un_le
tff(fact_6100_card__Un__le,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite1207074278014112911at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite1207074278014112911at_nat(A3)),finite1207074278014112911at_nat(B2)))) ).

% card_Un_le
tff(fact_6101_card__Un__le,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set168828743186997083at_nat(finite3771342082235030671at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set168828743186997083at_nat(finite3771342082235030671at_nat,A3)),aa_set168828743186997083at_nat(finite3771342082235030671at_nat,B2)))) ).

% card_Un_le
tff(fact_6102_periodic__finite__ex,axiom,
    ! [D: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X5: int,K2: int] :
            ( pp(aa_int_bool(P,X5))
          <=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) )
       => ( ? [X_1: int] : pp(aa_int_bool(P,X_1))
        <=> ? [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D)))
              & pp(aa_int_bool(P,X3)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_6103_aset_I7_J,axiom,
    ! [D4: int,A3: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ! [X4: int] :
          ( ! [Xa4: int] :
              ( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),A3))
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4))) ) ) ) ).

% aset(7)
tff(fact_6104_aset_I5_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(T),A3))
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A3))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4)),T)) ) ) ) ) ).

% aset(5)
tff(fact_6105_aset_I4_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(T),A3))
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A3))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( X4 != T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4) != T ) ) ) ) ) ).

% aset(4)
tff(fact_6106_aset_I3_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A3))
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A3))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( X4 = T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4) = T ) ) ) ) ) ).

% aset(3)
tff(fact_6107_bset_I7_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(T),B2))
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B2))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4))) ) ) ) ) ).

% bset(7)
tff(fact_6108_bset_I5_J,axiom,
    ! [D4: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ! [X4: int] :
          ( ! [Xa4: int] :
              ( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),B2))
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4)),T)) ) ) ) ).

% bset(5)
tff(fact_6109_bset_I4_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(T),B2))
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B2))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( X4 != T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4) != T ) ) ) ) ) ).

% bset(4)
tff(fact_6110_bset_I3_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B2))
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B2))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( X4 = T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4) = T ) ) ) ) ) ).

% bset(3)
tff(fact_6111_aset_I8_J,axiom,
    ! [D4: int,A3: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ! [X4: int] :
          ( ! [Xa4: int] :
              ( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),A3))
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4))) ) ) ) ).

% aset(8)
tff(fact_6112_aset_I6_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A3))
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A3))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4)),T)) ) ) ) ) ).

% aset(6)
tff(fact_6113_bset_I8_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B2))
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B2))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4))) ) ) ) ) ).

% bset(8)
tff(fact_6114_bset_I6_J,axiom,
    ! [D4: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ! [X4: int] :
          ( ! [Xa4: int] :
              ( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),B2))
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4)),T)) ) ) ) ).

% bset(6)
tff(fact_6115_cppi,axiom,
    ! [D4: int,P: fun_int_bool,P2: fun_int_bool,A3: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( ? [Z4: int] :
          ! [X5: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X5))
           => ( pp(aa_int_bool(P,X5))
            <=> pp(aa_int_bool(P2,X5)) ) )
       => ( ! [X5: int] :
              ( ! [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
                 => ! [Xb4: int] :
                      ( pp(aa_set_int_bool(member_int(Xb4),A3))
                     => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb4),Xa) ) ) )
             => ( pp(aa_int_bool(P,X5))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4))) ) )
         => ( ! [X5: int,K2: int] :
                ( pp(aa_int_bool(P2,X5))
              <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4)))) )
           => ( ? [X_1: int] : pp(aa_int_bool(P,X_1))
            <=> ( ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D4)))
                    & pp(aa_int_bool(P2,X3)) )
                | ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D4)))
                    & ? [Xa3: int] :
                        ( pp(aa_set_int_bool(member_int(Xa3),A3))
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X3))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_6116_cpmi,axiom,
    ! [D4: int,P: fun_int_bool,P2: fun_int_bool,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( ? [Z4: int] :
          ! [X5: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z4))
           => ( pp(aa_int_bool(P,X5))
            <=> pp(aa_int_bool(P2,X5)) ) )
       => ( ! [X5: int] :
              ( ! [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
                 => ! [Xb4: int] :
                      ( pp(aa_set_int_bool(member_int(Xb4),B2))
                     => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb4),Xa) ) ) )
             => ( pp(aa_int_bool(P,X5))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4))) ) )
         => ( ! [X5: int,K2: int] :
                ( pp(aa_int_bool(P2,X5))
              <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4)))) )
           => ( ? [X_1: int] : pp(aa_int_bool(P,X_1))
            <=> ( ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D4)))
                    & pp(aa_int_bool(P2,X3)) )
                | ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D4)))
                    & ? [Xa3: int] :
                        ( pp(aa_set_int_bool(member_int(Xa3),B2))
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X3))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_6117_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_6118_VEBT__internal_Oinsert_H_Osimps_I1_J,axiom,
    ! [A2: bool,B3: bool,X: nat] : vEBT_VEBT_insert(vEBT_Leaf(A2,B3),X) = vEBT_vebt_insert(vEBT_Leaf(A2,B3),X) ).

% VEBT_internal.insert'.simps(1)
tff(fact_6119_insert_H__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( vEBT_set_vebt(vEBT_VEBT_insert(T,X)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vEBT_set_vebt(T)),insert_nat(X,bot_bot_set_nat))),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))) ) ) ).

% insert'_correct
tff(fact_6120_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),N))
     => ( ( N = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),Mi))
         => ( ( Ma != Mi )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ).

% nested_mint
tff(fact_6121_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),X)) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_6122_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
        & ( ( X = Mi )
          | ( X = Ma )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
            & pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ).

% member_inv
tff(fact_6123_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),X))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_6124_both__member__options__ding,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(Info2,Deg,TreeList2,Summary)),X)) ) ) ) ).

% both_member_options_ding
tff(fact_6125_bit__split__inv,axiom,
    ! [X: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),D)),D) = X ).

% bit_split_inv
tff(fact_6126_Int__iff,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),inf_inf_set_real(A3,B2)))
    <=> ( pp(aa_set_real_bool(member_real(C),A3))
        & pp(aa_set_real_bool(member_real(C),B2)) ) ) ).

% Int_iff
tff(fact_6127_Int__iff,axiom,
    ! [C: bool,A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(C),inf_inf_set_bool(A3,B2)))
    <=> ( pp(aa_set_bool_bool(member_bool(C),A3))
        & pp(aa_set_bool_bool(member_bool(C),B2)) ) ) ).

% Int_iff
tff(fact_6128_Int__iff,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),inf_inf_set_set_nat(A3,B2)))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(C),A3))
        & pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ) ).

% Int_iff
tff(fact_6129_Int__iff,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)))
    <=> ( pp(aa_set_int_bool(member_int(C),A3))
        & pp(aa_set_int_bool(member_int(C),B2)) ) ) ).

% Int_iff
tff(fact_6130_Int__iff,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)))
    <=> ( pp(aa_set_nat_bool(member_nat(C),A3))
        & pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).

% Int_iff
tff(fact_6131_Int__iff,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)))
    <=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A3))
        & pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B2)) ) ) ).

% Int_iff
tff(fact_6132_IntI,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),A3))
     => ( pp(aa_set_real_bool(member_real(C),B2))
       => pp(aa_set_real_bool(member_real(C),inf_inf_set_real(A3,B2))) ) ) ).

% IntI
tff(fact_6133_IntI,axiom,
    ! [C: bool,A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(C),A3))
     => ( pp(aa_set_bool_bool(member_bool(C),B2))
       => pp(aa_set_bool_bool(member_bool(C),inf_inf_set_bool(A3,B2))) ) ) ).

% IntI
tff(fact_6134_IntI,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),A3))
     => ( pp(aa_set_set_nat_bool(member_set_nat(C),B2))
       => pp(aa_set_set_nat_bool(member_set_nat(C),inf_inf_set_set_nat(A3,B2))) ) ) ).

% IntI
tff(fact_6135_IntI,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),A3))
     => ( pp(aa_set_int_bool(member_int(C),B2))
       => pp(aa_set_int_bool(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))) ) ) ).

% IntI
tff(fact_6136_IntI,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),A3))
     => ( pp(aa_set_nat_bool(member_nat(C),B2))
       => pp(aa_set_nat_bool(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2))) ) ) ).

% IntI
tff(fact_6137_IntI,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A3))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B2))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2))) ) ) ).

% IntI
tff(fact_6138_low__def,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% low_def
tff(fact_6139_low__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),X)),N) = X ) ) ).

% low_inv
tff(fact_6140_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),X) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_zero_left
tff(fact_6141_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_real] : inf_inf_set_real(bot_bot_set_real,X) = bot_bot_set_real ).

% boolean_algebra.conj_zero_left
tff(fact_6142_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_bool] : inf_inf_set_bool(bot_bot_set_bool,X) = bot_bot_set_bool ).

% boolean_algebra.conj_zero_left
tff(fact_6143_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),X) = bot_bot_set_nat ).

% boolean_algebra.conj_zero_left
tff(fact_6144_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,bot_bot_set_int),X) = bot_bot_set_int ).

% boolean_algebra.conj_zero_left
tff(fact_6145_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_zero_right
tff(fact_6146_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_real] : inf_inf_set_real(X,bot_bot_set_real) = bot_bot_set_real ).

% boolean_algebra.conj_zero_right
tff(fact_6147_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_bool] : inf_inf_set_bool(X,bot_bot_set_bool) = bot_bot_set_bool ).

% boolean_algebra.conj_zero_right
tff(fact_6148_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),bot_bot_set_nat) = bot_bot_set_nat ).

% boolean_algebra.conj_zero_right
tff(fact_6149_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),bot_bot_set_int) = bot_bot_set_int ).

% boolean_algebra.conj_zero_right
tff(fact_6150_finite__Int,axiom,
    ! [F2: set_int,G2: set_int] :
      ( ( pp(aa_set_int_bool(finite_finite_int,F2))
        | pp(aa_set_int_bool(finite_finite_int,G2)) )
     => pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,F2),G2))) ) ).

% finite_Int
tff(fact_6151_finite__Int,axiom,
    ! [F2: set_complex,G2: set_complex] :
      ( ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F2))
        | pp(aa_set_complex_bool(finite3207457112153483333omplex,G2)) )
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,inf_inf_set_complex(F2,G2))) ) ).

% finite_Int
tff(fact_6152_finite__Int,axiom,
    ! [F2: set_Extended_enat,G2: set_Extended_enat] :
      ( ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F2))
        | pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,G2)) )
     => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,inf_in8357106775501769908d_enat(F2,G2))) ) ).

% finite_Int
tff(fact_6153_finite__Int,axiom,
    ! [F2: set_nat,G2: set_nat] :
      ( ( pp(aa_set_nat_bool(finite_finite_nat,F2))
        | pp(aa_set_nat_bool(finite_finite_nat,G2)) )
     => pp(aa_set_nat_bool(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,F2),G2))) ) ).

% finite_Int
tff(fact_6154_finite__Int,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
      ( ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F2))
        | pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,G2)) )
     => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,F2),G2))) ) ).

% finite_Int
tff(fact_6155_Int__subset__iff,axiom,
    ! [C2: set_nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C2),A3))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C2),B2)) ) ) ).

% Int_subset_iff
tff(fact_6156_Int__subset__iff,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)))
    <=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(C2),A3))
        & pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(C2),B2)) ) ) ).

% Int_subset_iff
tff(fact_6157_Int__subset__iff,axiom,
    ! [C2: set_int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),A3))
        & pp(aa_set_int_bool(ord_less_eq_set_int(C2),B2)) ) ) ).

% Int_subset_iff
tff(fact_6158_Int__insert__right__if1,axiom,
    ! [A2: real,A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(A2),A3))
     => ( inf_inf_set_real(A3,insert_real(A2,B2)) = insert_real(A2,inf_inf_set_real(A3,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_6159_Int__insert__right__if1,axiom,
    ! [A2: bool,A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(A2),A3))
     => ( inf_inf_set_bool(A3,insert_bool(A2,B2)) = insert_bool(A2,inf_inf_set_bool(A3,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_6160_Int__insert__right__if1,axiom,
    ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A2),A3))
     => ( inf_inf_set_set_nat(A3,insert_set_nat(A2,B2)) = insert_set_nat(A2,inf_inf_set_set_nat(A3,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_6161_Int__insert__right__if1,axiom,
    ! [A2: int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(A2),A3))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),insert_int(A2,B2)) = insert_int(A2,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)) ) ) ).

% Int_insert_right_if1
tff(fact_6162_Int__insert__right__if1,axiom,
    ! [A2: nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A2),A3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),insert_nat(A2,B2)) = insert_nat(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)) ) ) ).

% Int_insert_right_if1
tff(fact_6163_Int__insert__right__if1,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A3))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),insert8211810215607154385at_nat(A2,B2)) = insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)) ) ) ).

% Int_insert_right_if1
tff(fact_6164_Int__insert__right__if0,axiom,
    ! [A2: real,A3: set_real,B2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(A2),A3))
     => ( inf_inf_set_real(A3,insert_real(A2,B2)) = inf_inf_set_real(A3,B2) ) ) ).

% Int_insert_right_if0
tff(fact_6165_Int__insert__right__if0,axiom,
    ! [A2: bool,A3: set_bool,B2: set_bool] :
      ( ~ pp(aa_set_bool_bool(member_bool(A2),A3))
     => ( inf_inf_set_bool(A3,insert_bool(A2,B2)) = inf_inf_set_bool(A3,B2) ) ) ).

% Int_insert_right_if0
tff(fact_6166_Int__insert__right__if0,axiom,
    ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(A2),A3))
     => ( inf_inf_set_set_nat(A3,insert_set_nat(A2,B2)) = inf_inf_set_set_nat(A3,B2) ) ) ).

% Int_insert_right_if0
tff(fact_6167_Int__insert__right__if0,axiom,
    ! [A2: int,A3: set_int,B2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(A2),A3))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),insert_int(A2,B2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2) ) ) ).

% Int_insert_right_if0
tff(fact_6168_Int__insert__right__if0,axiom,
    ! [A2: nat,A3: set_nat,B2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(A2),A3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),insert_nat(A2,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) ) ) ).

% Int_insert_right_if0
tff(fact_6169_Int__insert__right__if0,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A3))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),insert8211810215607154385at_nat(A2,B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) ) ) ).

% Int_insert_right_if0
tff(fact_6170_insert__inter__insert,axiom,
    ! [A2: real,A3: set_real,B2: set_real] : inf_inf_set_real(insert_real(A2,A3),insert_real(A2,B2)) = insert_real(A2,inf_inf_set_real(A3,B2)) ).

% insert_inter_insert
tff(fact_6171_insert__inter__insert,axiom,
    ! [A2: bool,A3: set_bool,B2: set_bool] : inf_inf_set_bool(insert_bool(A2,A3),insert_bool(A2,B2)) = insert_bool(A2,inf_inf_set_bool(A3,B2)) ).

% insert_inter_insert
tff(fact_6172_insert__inter__insert,axiom,
    ! [A2: int,A3: set_int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A2,A3)),insert_int(A2,B2)) = insert_int(A2,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)) ).

% insert_inter_insert
tff(fact_6173_insert__inter__insert,axiom,
    ! [A2: nat,A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A2,A3)),insert_nat(A2,B2)) = insert_nat(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)) ).

% insert_inter_insert
tff(fact_6174_insert__inter__insert,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A2,A3)),insert8211810215607154385at_nat(A2,B2)) = insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)) ).

% insert_inter_insert
tff(fact_6175_Int__insert__left__if1,axiom,
    ! [A2: real,C2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(A2),C2))
     => ( inf_inf_set_real(insert_real(A2,B2),C2) = insert_real(A2,inf_inf_set_real(B2,C2)) ) ) ).

% Int_insert_left_if1
tff(fact_6176_Int__insert__left__if1,axiom,
    ! [A2: bool,C2: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(A2),C2))
     => ( inf_inf_set_bool(insert_bool(A2,B2),C2) = insert_bool(A2,inf_inf_set_bool(B2,C2)) ) ) ).

% Int_insert_left_if1
tff(fact_6177_Int__insert__left__if1,axiom,
    ! [A2: set_nat,C2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A2),C2))
     => ( inf_inf_set_set_nat(insert_set_nat(A2,B2),C2) = insert_set_nat(A2,inf_inf_set_set_nat(B2,C2)) ) ) ).

% Int_insert_left_if1
tff(fact_6178_Int__insert__left__if1,axiom,
    ! [A2: int,C2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(A2),C2))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A2,B2)),C2) = insert_int(A2,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_6179_Int__insert__left__if1,axiom,
    ! [A2: nat,C2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A2),C2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A2,B2)),C2) = insert_nat(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_6180_Int__insert__left__if1,axiom,
    ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),C2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A2,B2)),C2) = insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_6181_Int__insert__left__if0,axiom,
    ! [A2: real,C2: set_real,B2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(A2),C2))
     => ( inf_inf_set_real(insert_real(A2,B2),C2) = inf_inf_set_real(B2,C2) ) ) ).

% Int_insert_left_if0
tff(fact_6182_Int__insert__left__if0,axiom,
    ! [A2: bool,C2: set_bool,B2: set_bool] :
      ( ~ pp(aa_set_bool_bool(member_bool(A2),C2))
     => ( inf_inf_set_bool(insert_bool(A2,B2),C2) = inf_inf_set_bool(B2,C2) ) ) ).

% Int_insert_left_if0
tff(fact_6183_Int__insert__left__if0,axiom,
    ! [A2: set_nat,C2: set_set_nat,B2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(A2),C2))
     => ( inf_inf_set_set_nat(insert_set_nat(A2,B2),C2) = inf_inf_set_set_nat(B2,C2) ) ) ).

% Int_insert_left_if0
tff(fact_6184_Int__insert__left__if0,axiom,
    ! [A2: int,C2: set_int,B2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(A2),C2))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A2,B2)),C2) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),C2) ) ) ).

% Int_insert_left_if0
tff(fact_6185_Int__insert__left__if0,axiom,
    ! [A2: nat,C2: set_nat,B2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(A2),C2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A2,B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2) ) ) ).

% Int_insert_left_if0
tff(fact_6186_Int__insert__left__if0,axiom,
    ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),C2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A2,B2)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2) ) ) ).

% Int_insert_left_if0
tff(fact_6187_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T2)),S) = S ).

% Un_Int_eq(1)
tff(fact_6188_Un__Int__eq_I1_J,axiom,
    ! [S: set_nat,T2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T2)),S) = S ).

% Un_Int_eq(1)
tff(fact_6189_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,S),T2)),S) = S ).

% Un_Int_eq(1)
tff(fact_6190_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T2)),S) = S ).

% Un_Int_eq(1)
tff(fact_6191_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T2)),T2) = T2 ).

% Un_Int_eq(2)
tff(fact_6192_Un__Int__eq_I2_J,axiom,
    ! [S: set_nat,T2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T2)),T2) = T2 ).

% Un_Int_eq(2)
tff(fact_6193_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,S),T2)),T2) = T2 ).

% Un_Int_eq(2)
tff(fact_6194_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T2)),T2) = T2 ).

% Un_Int_eq(2)
tff(fact_6195_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T2)) = S ).

% Un_Int_eq(3)
tff(fact_6196_Un__Int__eq_I3_J,axiom,
    ! [S: set_nat,T2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T2)) = S ).

% Un_Int_eq(3)
tff(fact_6197_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,S),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,S),T2)) = S ).

% Un_Int_eq(3)
tff(fact_6198_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T2)) = S ).

% Un_Int_eq(3)
tff(fact_6199_Un__Int__eq_I4_J,axiom,
    ! [T2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,T2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T2)) = T2 ).

% Un_Int_eq(4)
tff(fact_6200_Un__Int__eq_I4_J,axiom,
    ! [T2: set_nat,S: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,T2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T2)) = T2 ).

% Un_Int_eq(4)
tff(fact_6201_Un__Int__eq_I4_J,axiom,
    ! [T2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,T2),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,S),T2)) = T2 ).

% Un_Int_eq(4)
tff(fact_6202_Un__Int__eq_I4_J,axiom,
    ! [T2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,T2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T2)) = T2 ).

% Un_Int_eq(4)
tff(fact_6203_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T2)),S) = S ).

% Int_Un_eq(1)
tff(fact_6204_Int__Un__eq_I1_J,axiom,
    ! [S: set_nat,T2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T2)),S) = S ).

% Int_Un_eq(1)
tff(fact_6205_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,S),T2)),S) = S ).

% Int_Un_eq(1)
tff(fact_6206_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),T2)),S) = S ).

% Int_Un_eq(1)
tff(fact_6207_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T2)),T2) = T2 ).

% Int_Un_eq(2)
tff(fact_6208_Int__Un__eq_I2_J,axiom,
    ! [S: set_nat,T2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T2)),T2) = T2 ).

% Int_Un_eq(2)
tff(fact_6209_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,S),T2)),T2) = T2 ).

% Int_Un_eq(2)
tff(fact_6210_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),T2)),T2) = T2 ).

% Int_Un_eq(2)
tff(fact_6211_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T2)) = S ).

% Int_Un_eq(3)
tff(fact_6212_Int__Un__eq_I3_J,axiom,
    ! [S: set_nat,T2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T2)) = S ).

% Int_Un_eq(3)
tff(fact_6213_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,S),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,S),T2)) = S ).

% Int_Un_eq(3)
tff(fact_6214_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),T2)) = S ).

% Int_Un_eq(3)
tff(fact_6215_Int__Un__eq_I4_J,axiom,
    ! [T2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,T2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T2)) = T2 ).

% Int_Un_eq(4)
tff(fact_6216_Int__Un__eq_I4_J,axiom,
    ! [T2: set_nat,S: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,T2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T2)) = T2 ).

% Int_Un_eq(4)
tff(fact_6217_Int__Un__eq_I4_J,axiom,
    ! [T2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,T2),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,S),T2)) = T2 ).

% Int_Un_eq(4)
tff(fact_6218_Int__Un__eq_I4_J,axiom,
    ! [T2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,T2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),T2)) = T2 ).

% Int_Un_eq(4)
tff(fact_6219_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),Deg))
     => ( ( Mi != Ma )
       => ( aa_option_nat_nat(the_nat,vEBT_vebt_maxt(Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% summaxma
tff(fact_6220_inf__compl__bot__left1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = bot_bo2099793752762293965at_nat ).

% inf_compl_bot_left1
tff(fact_6221_inf__compl__bot__left1,axiom,
    ! [X: set_real,Y: set_real] : inf_inf_set_real(uminus612125837232591019t_real(X),inf_inf_set_real(X,Y)) = bot_bot_set_real ).

% inf_compl_bot_left1
tff(fact_6222_inf__compl__bot__left1,axiom,
    ! [X: set_bool,Y: set_bool] : inf_inf_set_bool(uminus5166143097222041326t_bool(X),inf_inf_set_bool(X,Y)) = bot_bot_set_bool ).

% inf_compl_bot_left1
tff(fact_6223_inf__compl__bot__left1,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = bot_bot_set_nat ).

% inf_compl_bot_left1
tff(fact_6224_inf__compl__bot__left1,axiom,
    ! [X: set_int,Y: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(X)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)) = bot_bot_set_int ).

% inf_compl_bot_left1
tff(fact_6225_inf__compl__bot__left2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(X)),Y)) = bot_bo2099793752762293965at_nat ).

% inf_compl_bot_left2
tff(fact_6226_inf__compl__bot__left2,axiom,
    ! [X: set_real,Y: set_real] : inf_inf_set_real(X,inf_inf_set_real(uminus612125837232591019t_real(X),Y)) = bot_bot_set_real ).

% inf_compl_bot_left2
tff(fact_6227_inf__compl__bot__left2,axiom,
    ! [X: set_bool,Y: set_bool] : inf_inf_set_bool(X,inf_inf_set_bool(uminus5166143097222041326t_bool(X),Y)) = bot_bot_set_bool ).

% inf_compl_bot_left2
tff(fact_6228_inf__compl__bot__left2,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(X)),Y)) = bot_bot_set_nat ).

% inf_compl_bot_left2
tff(fact_6229_inf__compl__bot__left2,axiom,
    ! [X: set_int,Y: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(X)),Y)) = bot_bot_set_int ).

% inf_compl_bot_left2
tff(fact_6230_inf__compl__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),uminus6524753893492686040at_nat(X))) = bot_bo2099793752762293965at_nat ).

% inf_compl_bot_right
tff(fact_6231_inf__compl__bot__right,axiom,
    ! [X: set_real,Y: set_real] : inf_inf_set_real(X,inf_inf_set_real(Y,uminus612125837232591019t_real(X))) = bot_bot_set_real ).

% inf_compl_bot_right
tff(fact_6232_inf__compl__bot__right,axiom,
    ! [X: set_bool,Y: set_bool] : inf_inf_set_bool(X,inf_inf_set_bool(Y,uminus5166143097222041326t_bool(X))) = bot_bot_set_bool ).

% inf_compl_bot_right
tff(fact_6233_inf__compl__bot__right,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),uminus5710092332889474511et_nat(X))) = bot_bot_set_nat ).

% inf_compl_bot_right
tff(fact_6234_inf__compl__bot__right,axiom,
    ! [X: set_int,Y: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Y),uminus1532241313380277803et_int(X))) = bot_bot_set_int ).

% inf_compl_bot_right
tff(fact_6235_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(X)),X) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_cancel_left
tff(fact_6236_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_real] : inf_inf_set_real(uminus612125837232591019t_real(X),X) = bot_bot_set_real ).

% boolean_algebra.conj_cancel_left
tff(fact_6237_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_bool] : inf_inf_set_bool(uminus5166143097222041326t_bool(X),X) = bot_bot_set_bool ).

% boolean_algebra.conj_cancel_left
tff(fact_6238_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(X)),X) = bot_bot_set_nat ).

% boolean_algebra.conj_cancel_left
tff(fact_6239_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(X)),X) = bot_bot_set_int ).

% boolean_algebra.conj_cancel_left
tff(fact_6240_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),uminus6524753893492686040at_nat(X)) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_cancel_right
tff(fact_6241_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_real] : inf_inf_set_real(X,uminus612125837232591019t_real(X)) = bot_bot_set_real ).

% boolean_algebra.conj_cancel_right
tff(fact_6242_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_bool] : inf_inf_set_bool(X,uminus5166143097222041326t_bool(X)) = bot_bot_set_bool ).

% boolean_algebra.conj_cancel_right
tff(fact_6243_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),uminus5710092332889474511et_nat(X)) = bot_bot_set_nat ).

% boolean_algebra.conj_cancel_right
tff(fact_6244_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),uminus1532241313380277803et_int(X)) = bot_bot_set_int ).

% boolean_algebra.conj_cancel_right
tff(fact_6245_disjoint__insert_I2_J,axiom,
    ! [A3: set_set_nat,B3: set_nat,B2: set_set_nat] :
      ( ( bot_bot_set_set_nat = inf_inf_set_set_nat(A3,insert_set_nat(B3,B2)) )
    <=> ( ~ pp(aa_set_set_nat_bool(member_set_nat(B3),A3))
        & ( bot_bot_set_set_nat = inf_inf_set_set_nat(A3,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_6246_disjoint__insert_I2_J,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),insert8211810215607154385at_nat(B3,B2)) )
    <=> ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B3),A3))
        & ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_6247_disjoint__insert_I2_J,axiom,
    ! [A3: set_real,B3: real,B2: set_real] :
      ( ( bot_bot_set_real = inf_inf_set_real(A3,insert_real(B3,B2)) )
    <=> ( ~ pp(aa_set_real_bool(member_real(B3),A3))
        & ( bot_bot_set_real = inf_inf_set_real(A3,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_6248_disjoint__insert_I2_J,axiom,
    ! [A3: set_bool,B3: bool,B2: set_bool] :
      ( ( bot_bot_set_bool = inf_inf_set_bool(A3,insert_bool(B3,B2)) )
    <=> ( ~ pp(aa_set_bool_bool(member_bool(B3),A3))
        & ( bot_bot_set_bool = inf_inf_set_bool(A3,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_6249_disjoint__insert_I2_J,axiom,
    ! [A3: set_nat,B3: nat,B2: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),insert_nat(B3,B2)) )
    <=> ( ~ pp(aa_set_nat_bool(member_nat(B3),A3))
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_6250_disjoint__insert_I2_J,axiom,
    ! [A3: set_int,B3: int,B2: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),insert_int(B3,B2)) )
    <=> ( ~ pp(aa_set_int_bool(member_int(B3),A3))
        & ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_6251_disjoint__insert_I1_J,axiom,
    ! [B2: set_set_nat,A2: set_nat,A3: set_set_nat] :
      ( ( inf_inf_set_set_nat(B2,insert_set_nat(A2,A3)) = bot_bot_set_set_nat )
    <=> ( ~ pp(aa_set_set_nat_bool(member_set_nat(A2),B2))
        & ( inf_inf_set_set_nat(B2,A3) = bot_bot_set_set_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_6252_disjoint__insert_I1_J,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),insert8211810215607154385at_nat(A2,A3)) = bot_bo2099793752762293965at_nat )
    <=> ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),B2))
        & ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),A3) = bot_bo2099793752762293965at_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_6253_disjoint__insert_I1_J,axiom,
    ! [B2: set_real,A2: real,A3: set_real] :
      ( ( inf_inf_set_real(B2,insert_real(A2,A3)) = bot_bot_set_real )
    <=> ( ~ pp(aa_set_real_bool(member_real(A2),B2))
        & ( inf_inf_set_real(B2,A3) = bot_bot_set_real ) ) ) ).

% disjoint_insert(1)
tff(fact_6254_disjoint__insert_I1_J,axiom,
    ! [B2: set_bool,A2: bool,A3: set_bool] :
      ( ( inf_inf_set_bool(B2,insert_bool(A2,A3)) = bot_bot_set_bool )
    <=> ( ~ pp(aa_set_bool_bool(member_bool(A2),B2))
        & ( inf_inf_set_bool(B2,A3) = bot_bot_set_bool ) ) ) ).

% disjoint_insert(1)
tff(fact_6255_disjoint__insert_I1_J,axiom,
    ! [B2: set_nat,A2: nat,A3: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),insert_nat(A2,A3)) = bot_bot_set_nat )
    <=> ( ~ pp(aa_set_nat_bool(member_nat(A2),B2))
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),A3) = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_6256_disjoint__insert_I1_J,axiom,
    ! [B2: set_int,A2: int,A3: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),insert_int(A2,A3)) = bot_bot_set_int )
    <=> ( ~ pp(aa_set_int_bool(member_int(A2),B2))
        & ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),A3) = bot_bot_set_int ) ) ) ).

% disjoint_insert(1)
tff(fact_6257_insert__disjoint_I2_J,axiom,
    ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( bot_bot_set_set_nat = inf_inf_set_set_nat(insert_set_nat(A2,A3),B2) )
    <=> ( ~ pp(aa_set_set_nat_bool(member_set_nat(A2),B2))
        & ( bot_bot_set_set_nat = inf_inf_set_set_nat(A3,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_6258_insert__disjoint_I2_J,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A2,A3)),B2) )
    <=> ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),B2))
        & ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_6259_insert__disjoint_I2_J,axiom,
    ! [A2: real,A3: set_real,B2: set_real] :
      ( ( bot_bot_set_real = inf_inf_set_real(insert_real(A2,A3),B2) )
    <=> ( ~ pp(aa_set_real_bool(member_real(A2),B2))
        & ( bot_bot_set_real = inf_inf_set_real(A3,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_6260_insert__disjoint_I2_J,axiom,
    ! [A2: bool,A3: set_bool,B2: set_bool] :
      ( ( bot_bot_set_bool = inf_inf_set_bool(insert_bool(A2,A3),B2) )
    <=> ( ~ pp(aa_set_bool_bool(member_bool(A2),B2))
        & ( bot_bot_set_bool = inf_inf_set_bool(A3,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_6261_insert__disjoint_I2_J,axiom,
    ! [A2: nat,A3: set_nat,B2: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A2,A3)),B2) )
    <=> ( ~ pp(aa_set_nat_bool(member_nat(A2),B2))
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_6262_insert__disjoint_I2_J,axiom,
    ! [A2: int,A3: set_int,B2: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A2,A3)),B2) )
    <=> ( ~ pp(aa_set_int_bool(member_int(A2),B2))
        & ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_6263_insert__disjoint_I1_J,axiom,
    ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( inf_inf_set_set_nat(insert_set_nat(A2,A3),B2) = bot_bot_set_set_nat )
    <=> ( ~ pp(aa_set_set_nat_bool(member_set_nat(A2),B2))
        & ( inf_inf_set_set_nat(A3,B2) = bot_bot_set_set_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_6264_insert__disjoint_I1_J,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A2,A3)),B2) = bot_bo2099793752762293965at_nat )
    <=> ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),B2))
        & ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) = bot_bo2099793752762293965at_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_6265_insert__disjoint_I1_J,axiom,
    ! [A2: real,A3: set_real,B2: set_real] :
      ( ( inf_inf_set_real(insert_real(A2,A3),B2) = bot_bot_set_real )
    <=> ( ~ pp(aa_set_real_bool(member_real(A2),B2))
        & ( inf_inf_set_real(A3,B2) = bot_bot_set_real ) ) ) ).

% insert_disjoint(1)
tff(fact_6266_insert__disjoint_I1_J,axiom,
    ! [A2: bool,A3: set_bool,B2: set_bool] :
      ( ( inf_inf_set_bool(insert_bool(A2,A3),B2) = bot_bot_set_bool )
    <=> ( ~ pp(aa_set_bool_bool(member_bool(A2),B2))
        & ( inf_inf_set_bool(A3,B2) = bot_bot_set_bool ) ) ) ).

% insert_disjoint(1)
tff(fact_6267_insert__disjoint_I1_J,axiom,
    ! [A2: nat,A3: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A2,A3)),B2) = bot_bot_set_nat )
    <=> ( ~ pp(aa_set_nat_bool(member_nat(A2),B2))
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_6268_insert__disjoint_I1_J,axiom,
    ! [A2: int,A3: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A2,A3)),B2) = bot_bot_set_int )
    <=> ( ~ pp(aa_set_int_bool(member_int(A2),B2))
        & ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2) = bot_bot_set_int ) ) ) ).

% insert_disjoint(1)
tff(fact_6269_Diff__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),A3)) = bot_bo2099793752762293965at_nat ).

% Diff_disjoint
tff(fact_6270_Diff__disjoint,axiom,
    ! [A3: set_real,B2: set_real] : inf_inf_set_real(A3,minus_minus_set_real(B2,A3)) = bot_bot_set_real ).

% Diff_disjoint
tff(fact_6271_Diff__disjoint,axiom,
    ! [A3: set_bool,B2: set_bool] : inf_inf_set_bool(A3,minus_minus_set_bool(B2,A3)) = bot_bot_set_bool ).

% Diff_disjoint
tff(fact_6272_Diff__disjoint,axiom,
    ! [A3: set_int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B2),A3)) = bot_bot_set_int ).

% Diff_disjoint
tff(fact_6273_Diff__disjoint,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A3)) = bot_bot_set_nat ).

% Diff_disjoint
tff(fact_6274_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( some_nat(aa_option_nat_nat(the_nat,Option)) = Option ) ) ).

% option.collapse
tff(fact_6275_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_6276_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( some_num(the_num(Option)) = Option ) ) ).

% option.collapse
tff(fact_6277_Compl__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),uminus6524753893492686040at_nat(A3)) = bot_bo2099793752762293965at_nat ).

% Compl_disjoint
tff(fact_6278_Compl__disjoint,axiom,
    ! [A3: set_real] : inf_inf_set_real(A3,uminus612125837232591019t_real(A3)) = bot_bot_set_real ).

% Compl_disjoint
tff(fact_6279_Compl__disjoint,axiom,
    ! [A3: set_bool] : inf_inf_set_bool(A3,uminus5166143097222041326t_bool(A3)) = bot_bot_set_bool ).

% Compl_disjoint
tff(fact_6280_Compl__disjoint,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),uminus5710092332889474511et_nat(A3)) = bot_bot_set_nat ).

% Compl_disjoint
tff(fact_6281_Compl__disjoint,axiom,
    ! [A3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),uminus1532241313380277803et_int(A3)) = bot_bot_set_int ).

% Compl_disjoint
tff(fact_6282_Compl__disjoint2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(A3)),A3) = bot_bo2099793752762293965at_nat ).

% Compl_disjoint2
tff(fact_6283_Compl__disjoint2,axiom,
    ! [A3: set_real] : inf_inf_set_real(uminus612125837232591019t_real(A3),A3) = bot_bot_set_real ).

% Compl_disjoint2
tff(fact_6284_Compl__disjoint2,axiom,
    ! [A3: set_bool] : inf_inf_set_bool(uminus5166143097222041326t_bool(A3),A3) = bot_bot_set_bool ).

% Compl_disjoint2
tff(fact_6285_Compl__disjoint2,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(A3)),A3) = bot_bot_set_nat ).

% Compl_disjoint2
tff(fact_6286_Compl__disjoint2,axiom,
    ! [A3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(A3)),A3) = bot_bot_set_int ).

% Compl_disjoint2
tff(fact_6287_Diff__Compl,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),uminus6524753893492686040at_nat(B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) ).

% Diff_Compl
tff(fact_6288_Diff__Compl,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),uminus5710092332889474511et_nat(B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) ).

% Diff_Compl
tff(fact_6289_Int__left__commute,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),C2)) ).

% Int_left_commute
tff(fact_6290_Int__left__commute,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),C2)) ).

% Int_left_commute
tff(fact_6291_Int__left__absorb,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) ).

% Int_left_absorb
tff(fact_6292_Int__left__absorb,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) ).

% Int_left_absorb
tff(fact_6293_Int__commute,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),A3) ).

% Int_commute
tff(fact_6294_Int__commute,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),A3) ).

% Int_commute
tff(fact_6295_Int__absorb,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),A3) = A3 ).

% Int_absorb
tff(fact_6296_Int__absorb,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),A3) = A3 ).

% Int_absorb
tff(fact_6297_Int__assoc,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) ).

% Int_assoc
tff(fact_6298_Int__assoc,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) ).

% Int_assoc
tff(fact_6299_IntD2,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),inf_inf_set_real(A3,B2)))
     => pp(aa_set_real_bool(member_real(C),B2)) ) ).

% IntD2
tff(fact_6300_IntD2,axiom,
    ! [C: bool,A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(C),inf_inf_set_bool(A3,B2)))
     => pp(aa_set_bool_bool(member_bool(C),B2)) ) ).

% IntD2
tff(fact_6301_IntD2,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),inf_inf_set_set_nat(A3,B2)))
     => pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ).

% IntD2
tff(fact_6302_IntD2,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)))
     => pp(aa_set_int_bool(member_int(C),B2)) ) ).

% IntD2
tff(fact_6303_IntD2,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)))
     => pp(aa_set_nat_bool(member_nat(C),B2)) ) ).

% IntD2
tff(fact_6304_IntD2,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B2)) ) ).

% IntD2
tff(fact_6305_IntD1,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),inf_inf_set_real(A3,B2)))
     => pp(aa_set_real_bool(member_real(C),A3)) ) ).

% IntD1
tff(fact_6306_IntD1,axiom,
    ! [C: bool,A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(C),inf_inf_set_bool(A3,B2)))
     => pp(aa_set_bool_bool(member_bool(C),A3)) ) ).

% IntD1
tff(fact_6307_IntD1,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),inf_inf_set_set_nat(A3,B2)))
     => pp(aa_set_set_nat_bool(member_set_nat(C),A3)) ) ).

% IntD1
tff(fact_6308_IntD1,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)))
     => pp(aa_set_int_bool(member_int(C),A3)) ) ).

% IntD1
tff(fact_6309_IntD1,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)))
     => pp(aa_set_nat_bool(member_nat(C),A3)) ) ).

% IntD1
tff(fact_6310_IntD1,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A3)) ) ).

% IntD1
tff(fact_6311_IntE,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),inf_inf_set_real(A3,B2)))
     => ~ ( pp(aa_set_real_bool(member_real(C),A3))
         => ~ pp(aa_set_real_bool(member_real(C),B2)) ) ) ).

% IntE
tff(fact_6312_IntE,axiom,
    ! [C: bool,A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(C),inf_inf_set_bool(A3,B2)))
     => ~ ( pp(aa_set_bool_bool(member_bool(C),A3))
         => ~ pp(aa_set_bool_bool(member_bool(C),B2)) ) ) ).

% IntE
tff(fact_6313_IntE,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),inf_inf_set_set_nat(A3,B2)))
     => ~ ( pp(aa_set_set_nat_bool(member_set_nat(C),A3))
         => ~ pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ) ).

% IntE
tff(fact_6314_IntE,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)))
     => ~ ( pp(aa_set_int_bool(member_int(C),A3))
         => ~ pp(aa_set_int_bool(member_int(C),B2)) ) ) ).

% IntE
tff(fact_6315_IntE,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)))
     => ~ ( pp(aa_set_nat_bool(member_nat(C),A3))
         => ~ pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).

% IntE
tff(fact_6316_IntE,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)))
     => ~ ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A3))
         => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B2)) ) ) ).

% IntE
tff(fact_6317_Int__emptyI,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( ! [X5: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X5),A3))
         => ~ pp(aa_set_set_nat_bool(member_set_nat(X5),B2)) )
     => ( inf_inf_set_set_nat(A3,B2) = bot_bot_set_set_nat ) ) ).

% Int_emptyI
tff(fact_6318_Int__emptyI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ! [X5: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),A3))
         => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),B2)) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
tff(fact_6319_Int__emptyI,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),A3))
         => ~ pp(aa_set_real_bool(member_real(X5),B2)) )
     => ( inf_inf_set_real(A3,B2) = bot_bot_set_real ) ) ).

% Int_emptyI
tff(fact_6320_Int__emptyI,axiom,
    ! [A3: set_bool,B2: set_bool] :
      ( ! [X5: bool] :
          ( pp(aa_set_bool_bool(member_bool(X5),A3))
         => ~ pp(aa_set_bool_bool(member_bool(X5),B2)) )
     => ( inf_inf_set_bool(A3,B2) = bot_bot_set_bool ) ) ).

% Int_emptyI
tff(fact_6321_Int__emptyI,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),A3))
         => ~ pp(aa_set_nat_bool(member_nat(X5),B2)) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) = bot_bot_set_nat ) ) ).

% Int_emptyI
tff(fact_6322_Int__emptyI,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),A3))
         => ~ pp(aa_set_int_bool(member_int(X5),B2)) )
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2) = bot_bot_set_int ) ) ).

% Int_emptyI
tff(fact_6323_disjoint__iff,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( ( inf_inf_set_set_nat(A3,B2) = bot_bot_set_set_nat )
    <=> ! [X3: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X3),A3))
         => ~ pp(aa_set_set_nat_bool(member_set_nat(X3),B2)) ) ) ).

% disjoint_iff
tff(fact_6324_disjoint__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) = bot_bo2099793752762293965at_nat )
    <=> ! [X3: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A3))
         => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),B2)) ) ) ).

% disjoint_iff
tff(fact_6325_disjoint__iff,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ( inf_inf_set_real(A3,B2) = bot_bot_set_real )
    <=> ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A3))
         => ~ pp(aa_set_real_bool(member_real(X3),B2)) ) ) ).

% disjoint_iff
tff(fact_6326_disjoint__iff,axiom,
    ! [A3: set_bool,B2: set_bool] :
      ( ( inf_inf_set_bool(A3,B2) = bot_bot_set_bool )
    <=> ! [X3: bool] :
          ( pp(aa_set_bool_bool(member_bool(X3),A3))
         => ~ pp(aa_set_bool_bool(member_bool(X3),B2)) ) ) ).

% disjoint_iff
tff(fact_6327_disjoint__iff,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) = bot_bot_set_nat )
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A3))
         => ~ pp(aa_set_nat_bool(member_nat(X3),B2)) ) ) ).

% disjoint_iff
tff(fact_6328_disjoint__iff,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2) = bot_bot_set_int )
    <=> ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A3))
         => ~ pp(aa_set_int_bool(member_int(X3),B2)) ) ) ).

% disjoint_iff
tff(fact_6329_Int__empty__left,axiom,
    ! [B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),B2) = bot_bo2099793752762293965at_nat ).

% Int_empty_left
tff(fact_6330_Int__empty__left,axiom,
    ! [B2: set_real] : inf_inf_set_real(bot_bot_set_real,B2) = bot_bot_set_real ).

% Int_empty_left
tff(fact_6331_Int__empty__left,axiom,
    ! [B2: set_bool] : inf_inf_set_bool(bot_bot_set_bool,B2) = bot_bot_set_bool ).

% Int_empty_left
tff(fact_6332_Int__empty__left,axiom,
    ! [B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),B2) = bot_bot_set_nat ).

% Int_empty_left
tff(fact_6333_Int__empty__left,axiom,
    ! [B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,bot_bot_set_int),B2) = bot_bot_set_int ).

% Int_empty_left
tff(fact_6334_Int__empty__right,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% Int_empty_right
tff(fact_6335_Int__empty__right,axiom,
    ! [A3: set_real] : inf_inf_set_real(A3,bot_bot_set_real) = bot_bot_set_real ).

% Int_empty_right
tff(fact_6336_Int__empty__right,axiom,
    ! [A3: set_bool] : inf_inf_set_bool(A3,bot_bot_set_bool) = bot_bot_set_bool ).

% Int_empty_right
tff(fact_6337_Int__empty__right,axiom,
    ! [A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),bot_bot_set_nat) = bot_bot_set_nat ).

% Int_empty_right
tff(fact_6338_Int__empty__right,axiom,
    ! [A3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),bot_bot_set_int) = bot_bot_set_int ).

% Int_empty_right
tff(fact_6339_disjoint__iff__not__equal,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) = bot_bo2099793752762293965at_nat )
    <=> ! [X3: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A3))
         => ! [Xa3: product_prod_nat_nat] :
              ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Xa3),B2))
             => ( X3 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_6340_disjoint__iff__not__equal,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ( inf_inf_set_real(A3,B2) = bot_bot_set_real )
    <=> ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A3))
         => ! [Xa3: real] :
              ( pp(aa_set_real_bool(member_real(Xa3),B2))
             => ( X3 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_6341_disjoint__iff__not__equal,axiom,
    ! [A3: set_bool,B2: set_bool] :
      ( ( inf_inf_set_bool(A3,B2) = bot_bot_set_bool )
    <=> ! [X3: bool] :
          ( pp(aa_set_bool_bool(member_bool(X3),A3))
         => ! [Xa3: bool] :
              ( pp(aa_set_bool_bool(member_bool(Xa3),B2))
             => ( pp(X3)
              <=> ~ pp(Xa3) ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_6342_disjoint__iff__not__equal,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) = bot_bot_set_nat )
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A3))
         => ! [Xa3: nat] :
              ( pp(aa_set_nat_bool(member_nat(Xa3),B2))
             => ( X3 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_6343_disjoint__iff__not__equal,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2) = bot_bot_set_int )
    <=> ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A3))
         => ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),B2))
             => ( X3 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_6344_Int__mono,axiom,
    ! [A3: set_nat,C2: set_nat,B2: set_nat,D4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),C2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),D4))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),D4))) ) ) ).

% Int_mono
tff(fact_6345_Int__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,D4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),C2))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(B2),D4))
       => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),D4))) ) ) ).

% Int_mono
tff(fact_6346_Int__mono,axiom,
    ! [A3: set_int,C2: set_int,B2: set_int,D4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),C2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),D4))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,C2),D4))) ) ) ).

% Int_mono
tff(fact_6347_Int__lower1,axiom,
    ! [A3: set_nat,B2: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)),A3)) ).

% Int_lower1
tff(fact_6348_Int__lower1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)),A3)) ).

% Int_lower1
tff(fact_6349_Int__lower1,axiom,
    ! [A3: set_int,B2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)),A3)) ).

% Int_lower1
tff(fact_6350_Int__lower2,axiom,
    ! [A3: set_nat,B2: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)),B2)) ).

% Int_lower2
tff(fact_6351_Int__lower2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)),B2)) ).

% Int_lower2
tff(fact_6352_Int__lower2,axiom,
    ! [A3: set_int,B2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)),B2)) ).

% Int_lower2
tff(fact_6353_Int__absorb1,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) = B2 ) ) ).

% Int_absorb1
tff(fact_6354_Int__absorb1,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(B2),A3))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) = B2 ) ) ).

% Int_absorb1
tff(fact_6355_Int__absorb1,axiom,
    ! [B2: set_int,A3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A3))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2) = B2 ) ) ).

% Int_absorb1
tff(fact_6356_Int__absorb2,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) = A3 ) ) ).

% Int_absorb2
tff(fact_6357_Int__absorb2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),B2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) = A3 ) ) ).

% Int_absorb2
tff(fact_6358_Int__absorb2,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2) = A3 ) ) ).

% Int_absorb2
tff(fact_6359_Int__greatest,axiom,
    ! [C2: set_nat,A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C2),A3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C2),B2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2))) ) ) ).

% Int_greatest
tff(fact_6360_Int__greatest,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(C2),A3))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(C2),B2))
       => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2))) ) ) ).

% Int_greatest
tff(fact_6361_Int__greatest,axiom,
    ! [C2: set_int,A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),A3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),B2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))) ) ) ).

% Int_greatest
tff(fact_6362_Int__Collect__mono,axiom,
    ! [A3: set_bool,B2: set_bool,P: fun_bool_bool,Q: fun_bool_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),B2))
     => ( ! [X5: bool] :
            ( pp(aa_set_bool_bool(member_bool(X5),A3))
           => ( pp(aa_bool_bool(P,X5))
             => pp(aa_bool_bool(Q,X5)) ) )
       => pp(aa_set_bool_bool(ord_less_eq_set_bool(inf_inf_set_bool(A3,collect_bool(P))),inf_inf_set_bool(B2,collect_bool(Q)))) ) ) ).

% Int_Collect_mono
tff(fact_6363_Int__Collect__mono,axiom,
    ! [A3: set_real,B2: set_real,P: fun_real_bool,Q: fun_real_bool] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2))
     => ( ! [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),A3))
           => ( pp(aa_real_bool(P,X5))
             => pp(aa_real_bool(Q,X5)) ) )
       => pp(aa_set_real_bool(ord_less_eq_set_real(inf_inf_set_real(A3,collect_real(P))),inf_inf_set_real(B2,collect_real(Q)))) ) ) ).

% Int_Collect_mono
tff(fact_6364_Int__Collect__mono,axiom,
    ! [A3: set_list_nat,B2: set_list_nat,P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ord_le6045566169113846134st_nat(A3,B2)
     => ( ! [X5: list_nat] :
            ( member_list_nat(X5,A3)
           => ( pp(aa_list_nat_bool(P,X5))
             => pp(aa_list_nat_bool(Q,X5)) ) )
       => ord_le6045566169113846134st_nat(inf_inf_set_list_nat(A3,collect_list_nat(P)),inf_inf_set_list_nat(B2,collect_list_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_6365_Int__Collect__mono,axiom,
    ! [A3: set_set_nat,B2: set_set_nat,P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),B2))
     => ( ! [X5: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X5),A3))
           => ( pp(aa_set_nat_bool(P,X5))
             => pp(aa_set_nat_bool(Q,X5)) ) )
       => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(inf_inf_set_set_nat(A3,collect_set_nat(P))),inf_inf_set_set_nat(B2,collect_set_nat(Q)))) ) ) ).

% Int_Collect_mono
tff(fact_6366_Int__Collect__mono,axiom,
    ! [A3: set_nat,B2: set_nat,P: fun_nat_bool,Q: fun_nat_bool] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
     => ( ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A3))
           => ( pp(aa_nat_bool(P,X5))
             => pp(aa_nat_bool(Q,X5)) ) )
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(P))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),collect_nat(Q)))) ) ) ).

% Int_Collect_mono
tff(fact_6367_Int__Collect__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,P: fun_Pr7757163693017731481t_bool,Q: fun_Pr7757163693017731481t_bool] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),B2))
     => ( ! [X5: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),A3))
           => ( pp(aa_Pro2232559814647311250t_bool(P,X5))
             => pp(aa_Pro2232559814647311250t_bool(Q,X5)) ) )
       => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),collec3392354462482085612at_nat(P))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),collec3392354462482085612at_nat(Q)))) ) ) ).

% Int_Collect_mono
tff(fact_6368_Int__Collect__mono,axiom,
    ! [A3: set_int,B2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
     => ( ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A3))
           => ( pp(aa_int_bool(P,X5))
             => pp(aa_int_bool(Q,X5)) ) )
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(P))),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),collect_int(Q)))) ) ) ).

% Int_Collect_mono
tff(fact_6369_Int__insert__right,axiom,
    ! [A2: real,A3: set_real,B2: set_real] :
      ( ( pp(aa_set_real_bool(member_real(A2),A3))
       => ( inf_inf_set_real(A3,insert_real(A2,B2)) = insert_real(A2,inf_inf_set_real(A3,B2)) ) )
      & ( ~ pp(aa_set_real_bool(member_real(A2),A3))
       => ( inf_inf_set_real(A3,insert_real(A2,B2)) = inf_inf_set_real(A3,B2) ) ) ) ).

% Int_insert_right
tff(fact_6370_Int__insert__right,axiom,
    ! [A2: bool,A3: set_bool,B2: set_bool] :
      ( ( pp(aa_set_bool_bool(member_bool(A2),A3))
       => ( inf_inf_set_bool(A3,insert_bool(A2,B2)) = insert_bool(A2,inf_inf_set_bool(A3,B2)) ) )
      & ( ~ pp(aa_set_bool_bool(member_bool(A2),A3))
       => ( inf_inf_set_bool(A3,insert_bool(A2,B2)) = inf_inf_set_bool(A3,B2) ) ) ) ).

% Int_insert_right
tff(fact_6371_Int__insert__right,axiom,
    ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( pp(aa_set_set_nat_bool(member_set_nat(A2),A3))
       => ( inf_inf_set_set_nat(A3,insert_set_nat(A2,B2)) = insert_set_nat(A2,inf_inf_set_set_nat(A3,B2)) ) )
      & ( ~ pp(aa_set_set_nat_bool(member_set_nat(A2),A3))
       => ( inf_inf_set_set_nat(A3,insert_set_nat(A2,B2)) = inf_inf_set_set_nat(A3,B2) ) ) ) ).

% Int_insert_right
tff(fact_6372_Int__insert__right,axiom,
    ! [A2: int,A3: set_int,B2: set_int] :
      ( ( pp(aa_set_int_bool(member_int(A2),A3))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),insert_int(A2,B2)) = insert_int(A2,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)) ) )
      & ( ~ pp(aa_set_int_bool(member_int(A2),A3))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),insert_int(A2,B2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2) ) ) ) ).

% Int_insert_right
tff(fact_6373_Int__insert__right,axiom,
    ! [A2: nat,A3: set_nat,B2: set_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(A2),A3))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),insert_nat(A2,B2)) = insert_nat(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(A2),A3))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),insert_nat(A2,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) ) ) ) ).

% Int_insert_right
tff(fact_6374_Int__insert__right,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A3))
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),insert8211810215607154385at_nat(A2,B2)) = insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)) ) )
      & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A3))
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),insert8211810215607154385at_nat(A2,B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) ) ) ) ).

% Int_insert_right
tff(fact_6375_Int__insert__left,axiom,
    ! [A2: real,C2: set_real,B2: set_real] :
      ( ( pp(aa_set_real_bool(member_real(A2),C2))
       => ( inf_inf_set_real(insert_real(A2,B2),C2) = insert_real(A2,inf_inf_set_real(B2,C2)) ) )
      & ( ~ pp(aa_set_real_bool(member_real(A2),C2))
       => ( inf_inf_set_real(insert_real(A2,B2),C2) = inf_inf_set_real(B2,C2) ) ) ) ).

% Int_insert_left
tff(fact_6376_Int__insert__left,axiom,
    ! [A2: bool,C2: set_bool,B2: set_bool] :
      ( ( pp(aa_set_bool_bool(member_bool(A2),C2))
       => ( inf_inf_set_bool(insert_bool(A2,B2),C2) = insert_bool(A2,inf_inf_set_bool(B2,C2)) ) )
      & ( ~ pp(aa_set_bool_bool(member_bool(A2),C2))
       => ( inf_inf_set_bool(insert_bool(A2,B2),C2) = inf_inf_set_bool(B2,C2) ) ) ) ).

% Int_insert_left
tff(fact_6377_Int__insert__left,axiom,
    ! [A2: set_nat,C2: set_set_nat,B2: set_set_nat] :
      ( ( pp(aa_set_set_nat_bool(member_set_nat(A2),C2))
       => ( inf_inf_set_set_nat(insert_set_nat(A2,B2),C2) = insert_set_nat(A2,inf_inf_set_set_nat(B2,C2)) ) )
      & ( ~ pp(aa_set_set_nat_bool(member_set_nat(A2),C2))
       => ( inf_inf_set_set_nat(insert_set_nat(A2,B2),C2) = inf_inf_set_set_nat(B2,C2) ) ) ) ).

% Int_insert_left
tff(fact_6378_Int__insert__left,axiom,
    ! [A2: int,C2: set_int,B2: set_int] :
      ( ( pp(aa_set_int_bool(member_int(A2),C2))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A2,B2)),C2) = insert_int(A2,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),C2)) ) )
      & ( ~ pp(aa_set_int_bool(member_int(A2),C2))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A2,B2)),C2) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),C2) ) ) ) ).

% Int_insert_left
tff(fact_6379_Int__insert__left,axiom,
    ! [A2: nat,C2: set_nat,B2: set_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(A2),C2))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A2,B2)),C2) = insert_nat(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(A2),C2))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A2,B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2) ) ) ) ).

% Int_insert_left
tff(fact_6380_Int__insert__left,axiom,
    ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),C2))
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A2,B2)),C2) = insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) ) )
      & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),C2))
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A2,B2)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2) ) ) ) ).

% Int_insert_left
tff(fact_6381_Un__Int__distrib2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)),A3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),A3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C2),A3)) ).

% Un_Int_distrib2
tff(fact_6382_Un__Int__distrib2,axiom,
    ! [B2: set_nat,C2: set_nat,A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)),A3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),A3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),A3)) ).

% Un_Int_distrib2
tff(fact_6383_Un__Int__distrib2,axiom,
    ! [B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,B2),C2)),A3) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B2),A3)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,C2),A3)) ).

% Un_Int_distrib2
tff(fact_6384_Un__Int__distrib2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B2),C2)),A3) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),A3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C2),A3)) ).

% Un_Int_distrib2
tff(fact_6385_Int__Un__distrib2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C2)),A3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),A3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A3)) ).

% Int_Un_distrib2
tff(fact_6386_Int__Un__distrib2,axiom,
    ! [B2: set_nat,C2: set_nat,A3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)),A3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),A3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A3)) ).

% Int_Un_distrib2
tff(fact_6387_Int__Un__distrib2,axiom,
    ! [B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B2),C2)),A3) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,B2),A3)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,C2),A3)) ).

% Int_Un_distrib2
tff(fact_6388_Int__Un__distrib2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2)),A3) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B2),A3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,C2),A3)) ).

% Int_Un_distrib2
tff(fact_6389_Un__Int__distrib,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),C2)) ).

% Un_Int_distrib
tff(fact_6390_Un__Int__distrib,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),C2)) ).

% Un_Int_distrib
tff(fact_6391_Un__Int__distrib,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,B2),C2)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),C2)) ).

% Un_Int_distrib
tff(fact_6392_Un__Int__distrib,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B2),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),C2)) ).

% Un_Int_distrib
tff(fact_6393_Int__Un__distrib,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),C2)) ).

% Int_Un_distrib
tff(fact_6394_Int__Un__distrib,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),C2)) ).

% Int_Un_distrib
tff(fact_6395_Int__Un__distrib,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B2),C2)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B2)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),C2)) ).

% Int_Un_distrib
tff(fact_6396_Int__Un__distrib,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),C2)) ).

% Int_Un_distrib
tff(fact_6397_Un__Int__crazy,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C2))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C2),A3)) ).

% Un_Int_crazy
tff(fact_6398_Un__Int__crazy,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),A3)) ).

% Un_Int_crazy
tff(fact_6399_Un__Int__crazy,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B2)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,B2),C2))),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,C2),A3)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B2),C2))),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,C2),A3)) ).

% Un_Int_crazy
tff(fact_6400_Un__Int__crazy,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B2),C2))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,C2),A3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C2),A3)) ).

% Un_Int_crazy
tff(fact_6401_Diff__Int__distrib2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),C2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) ).

% Diff_Int_distrib2
tff(fact_6402_Diff__Int__distrib2,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) ).

% Diff_Int_distrib2
tff(fact_6403_Diff__Int__distrib,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),B2)) ).

% Diff_Int_distrib
tff(fact_6404_Diff__Int__distrib,axiom,
    ! [C2: set_nat,A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),B2)) ).

% Diff_Int_distrib
tff(fact_6405_Diff__Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) ).

% Diff_Diff_Int
tff(fact_6406_Diff__Diff__Int,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) ).

% Diff_Diff_Int
tff(fact_6407_Diff__Int2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),C2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),C2)),B2) ).

% Diff_Int2
tff(fact_6408_Diff__Int2,axiom,
    ! [A3: set_nat,C2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),C2)),B2) ).

% Diff_Int2
tff(fact_6409_Int__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),C2)) ).

% Int_Diff
tff(fact_6410_Int__Diff,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),C2)) ).

% Int_Diff
tff(fact_6411_option_Osel,axiom,
    ! [X2: nat] : aa_option_nat_nat(the_nat,some_nat(X2)) = X2 ).

% option.sel
tff(fact_6412_option_Osel,axiom,
    ! [X2: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X2)) = X2 ).

% option.sel
tff(fact_6413_option_Osel,axiom,
    ! [X2: num] : the_num(some_num(X2)) = X2 ).

% option.sel
tff(fact_6414_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
      <=> ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( aa_option_nat_nat(the_nat,Option) = aa_option_nat_nat(the_nat,Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_6415_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_6416_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_6417_inf__cancel__left1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),A2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(X)),B3)) = bot_bo2099793752762293965at_nat ).

% inf_cancel_left1
tff(fact_6418_inf__cancel__left1,axiom,
    ! [X: set_real,A2: set_real,B3: set_real] : inf_inf_set_real(inf_inf_set_real(X,A2),inf_inf_set_real(uminus612125837232591019t_real(X),B3)) = bot_bot_set_real ).

% inf_cancel_left1
tff(fact_6419_inf__cancel__left1,axiom,
    ! [X: set_bool,A2: set_bool,B3: set_bool] : inf_inf_set_bool(inf_inf_set_bool(X,A2),inf_inf_set_bool(uminus5166143097222041326t_bool(X),B3)) = bot_bot_set_bool ).

% inf_cancel_left1
tff(fact_6420_inf__cancel__left1,axiom,
    ! [X: set_nat,A2: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(X)),B3)) = bot_bot_set_nat ).

% inf_cancel_left1
tff(fact_6421_inf__cancel__left1,axiom,
    ! [X: set_int,A2: set_int,B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),A2)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(X)),B3)) = bot_bot_set_int ).

% inf_cancel_left1
tff(fact_6422_inf__cancel__left2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(X)),A2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),B3)) = bot_bo2099793752762293965at_nat ).

% inf_cancel_left2
tff(fact_6423_inf__cancel__left2,axiom,
    ! [X: set_real,A2: set_real,B3: set_real] : inf_inf_set_real(inf_inf_set_real(uminus612125837232591019t_real(X),A2),inf_inf_set_real(X,B3)) = bot_bot_set_real ).

% inf_cancel_left2
tff(fact_6424_inf__cancel__left2,axiom,
    ! [X: set_bool,A2: set_bool,B3: set_bool] : inf_inf_set_bool(inf_inf_set_bool(uminus5166143097222041326t_bool(X),A2),inf_inf_set_bool(X,B3)) = bot_bot_set_bool ).

% inf_cancel_left2
tff(fact_6425_inf__cancel__left2,axiom,
    ! [X: set_nat,A2: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(X)),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),B3)) = bot_bot_set_nat ).

% inf_cancel_left2
tff(fact_6426_inf__cancel__left2,axiom,
    ! [X: set_int,A2: set_int,B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(X)),A2)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),B3)) = bot_bot_set_int ).

% inf_cancel_left2
tff(fact_6427_Diff__triv,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) = bot_bo2099793752762293965at_nat )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2) = A3 ) ) ).

% Diff_triv
tff(fact_6428_Diff__triv,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ( inf_inf_set_real(A3,B2) = bot_bot_set_real )
     => ( minus_minus_set_real(A3,B2) = A3 ) ) ).

% Diff_triv
tff(fact_6429_Diff__triv,axiom,
    ! [A3: set_bool,B2: set_bool] :
      ( ( inf_inf_set_bool(A3,B2) = bot_bot_set_bool )
     => ( minus_minus_set_bool(A3,B2) = A3 ) ) ).

% Diff_triv
tff(fact_6430_Diff__triv,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2) = bot_bot_set_int )
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2) = A3 ) ) ).

% Diff_triv
tff(fact_6431_Diff__triv,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) = bot_bot_set_nat )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2) = A3 ) ) ).

% Diff_triv
tff(fact_6432_Int__Diff__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2)) = bot_bo2099793752762293965at_nat ).

% Int_Diff_disjoint
tff(fact_6433_Int__Diff__disjoint,axiom,
    ! [A3: set_real,B2: set_real] : inf_inf_set_real(inf_inf_set_real(A3,B2),minus_minus_set_real(A3,B2)) = bot_bot_set_real ).

% Int_Diff_disjoint
tff(fact_6434_Int__Diff__disjoint,axiom,
    ! [A3: set_bool,B2: set_bool] : inf_inf_set_bool(inf_inf_set_bool(A3,B2),minus_minus_set_bool(A3,B2)) = bot_bot_set_bool ).

% Int_Diff_disjoint
tff(fact_6435_Int__Diff__disjoint,axiom,
    ! [A3: set_int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2)) = bot_bot_set_int ).

% Int_Diff_disjoint
tff(fact_6436_Int__Diff__disjoint,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)) = bot_bot_set_nat ).

% Int_Diff_disjoint
tff(fact_6437_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C2)) )
    <=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(C2),A3)) ) ).

% Un_Int_assoc_eq
tff(fact_6438_Un__Int__assoc__eq,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C2),A3)) ) ).

% Un_Int_assoc_eq
tff(fact_6439_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B2)),C2) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B2),C2)) )
    <=> ord_le3000389064537975527at_nat(C2,A3) ) ).

% Un_Int_assoc_eq
tff(fact_6440_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),B2)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2)) )
    <=> ord_le1268244103169919719at_nat(C2,A3) ) ).

% Un_Int_assoc_eq
tff(fact_6441_Un__Int__assoc__eq,axiom,
    ! [A3: set_int,B2: set_int,C2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)),C2) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B2),C2)) )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(C2),A3)) ) ).

% Un_Int_assoc_eq
tff(fact_6442_Un__Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)) = A3 ).

% Un_Diff_Int
tff(fact_6443_Un__Diff__Int,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),B2)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B2)) = A3 ).

% Un_Diff_Int
tff(fact_6444_Un__Diff__Int,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),B2)) = A3 ).

% Un_Diff_Int
tff(fact_6445_Un__Diff__Int,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)) = A3 ).

% Un_Diff_Int
tff(fact_6446_Int__Diff__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2)) = A3 ).

% Int_Diff_Un
tff(fact_6447_Int__Diff__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B2)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),B2)) = A3 ).

% Int_Diff_Un
tff(fact_6448_Int__Diff__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),B2)) = A3 ).

% Int_Diff_Un
tff(fact_6449_Int__Diff__Un,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)) = A3 ).

% Int_Diff_Un
tff(fact_6450_Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),C2)) ).

% Diff_Int
tff(fact_6451_Diff__Int,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,B2),C2)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),B2)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),C2)) ).

% Diff_Int
tff(fact_6452_Diff__Int,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B2),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),C2)) ).

% Diff_Int
tff(fact_6453_Diff__Int,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),C2)) ).

% Diff_Int
tff(fact_6454_Diff__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),C2)) ).

% Diff_Un
tff(fact_6455_Diff__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B2),C2)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),B2)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(minus_8321449233255521966at_nat,A3),C2)) ).

% Diff_Un
tff(fact_6456_Diff__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),B2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A3),C2)) ).

% Diff_Un
tff(fact_6457_Diff__Un,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),C2)) ).

% Diff_Un
tff(fact_6458_Compl__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat(A3)),uminus6524753893492686040at_nat(B2)) ).

% Compl_Int
tff(fact_6459_Compl__Int,axiom,
    ! [A3: set_nat,B2: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(A3)),uminus5710092332889474511et_nat(B2)) ).

% Compl_Int
tff(fact_6460_Compl__Int,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : uminus4384627049435823934at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B2)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,uminus4384627049435823934at_nat(A3)),uminus4384627049435823934at_nat(B2)) ).

% Compl_Int
tff(fact_6461_Compl__Int,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),B2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(A3)),uminus935396558254630718at_nat(B2)) ).

% Compl_Int
tff(fact_6462_Compl__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(A3)),uminus6524753893492686040at_nat(B2)) ).

% Compl_Un
tff(fact_6463_Compl__Un,axiom,
    ! [A3: set_nat,B2: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(A3)),uminus5710092332889474511et_nat(B2)) ).

% Compl_Un
tff(fact_6464_Compl__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : uminus4384627049435823934at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,uminus4384627049435823934at_nat(A3)),uminus4384627049435823934at_nat(B2)) ).

% Compl_Un
tff(fact_6465_Compl__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,uminus935396558254630718at_nat(A3)),uminus935396558254630718at_nat(B2)) ).

% Compl_Un
tff(fact_6466_Diff__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),uminus6524753893492686040at_nat(B2)) ).

% Diff_eq
tff(fact_6467_Diff__eq,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),uminus5710092332889474511et_nat(B2)) ).

% Diff_eq
tff(fact_6468_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = some_nat(aa_option_nat_nat(the_nat,Option)) ) ) ).

% option.exhaust_sel
tff(fact_6469_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_6470_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option = some_num(the_num(Option)) ) ) ).

% option.exhaust_sel
tff(fact_6471_inf__shunt,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = bot_bo2099793752762293965at_nat )
    <=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),uminus6524753893492686040at_nat(Y))) ) ).

% inf_shunt
tff(fact_6472_inf__shunt,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( inf_inf_set_real(X,Y) = bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(X),uminus612125837232591019t_real(Y))) ) ).

% inf_shunt
tff(fact_6473_inf__shunt,axiom,
    ! [X: set_bool,Y: set_bool] :
      ( ( inf_inf_set_bool(X,Y) = bot_bot_set_bool )
    <=> pp(aa_set_bool_bool(ord_less_eq_set_bool(X),uminus5166143097222041326t_bool(Y))) ) ).

% inf_shunt
tff(fact_6474_inf__shunt,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),uminus5710092332889474511et_nat(Y))) ) ).

% inf_shunt
tff(fact_6475_inf__shunt,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X),uminus1532241313380277803et_int(Y))) ) ).

% inf_shunt
tff(fact_6476_shunt1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Z))
    <=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat(Y)),Z))) ) ).

% shunt1
tff(fact_6477_shunt1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Z))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(Y)),Z))) ) ).

% shunt1
tff(fact_6478_shunt1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),Y),Z)
    <=> ord_le3000389064537975527at_nat(X,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,uminus4384627049435823934at_nat(Y)),Z)) ) ).

% shunt1
tff(fact_6479_shunt1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y),Z)
    <=> ord_le1268244103169919719at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(Y)),Z)) ) ).

% shunt1
tff(fact_6480_shunt1,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)),Z))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,uminus1532241313380277803et_int(Y)),Z))) ) ).

% shunt1
tff(fact_6481_shunt2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),uminus6524753893492686040at_nat(Y))),Z))
    <=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z))) ) ).

% shunt2
tff(fact_6482_shunt2,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),uminus5710092332889474511et_nat(Y))),Z))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z))) ) ).

% shunt2
tff(fact_6483_shunt2,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),uminus4384627049435823934at_nat(Y)),Z)
    <=> ord_le3000389064537975527at_nat(X,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),Z)) ) ).

% shunt2
tff(fact_6484_shunt2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),uminus935396558254630718at_nat(Y)),Z)
    <=> ord_le1268244103169919719at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) ) ).

% shunt2
tff(fact_6485_shunt2,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),uminus1532241313380277803et_int(Y))),Z))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Y),Z))) ) ).

% shunt2
tff(fact_6486_sup__neg__inf,axiom,
    ! [P3: set_Pr1261947904930325089at_nat,Q3: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(P3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Q3),R2)))
    <=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,P3),uminus6524753893492686040at_nat(Q3))),R2)) ) ).

% sup_neg_inf
tff(fact_6487_sup__neg__inf,axiom,
    ! [P3: set_nat,Q3: set_nat,R2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,P3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Q3),R2)))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,P3),uminus5710092332889474511et_nat(Q3))),R2)) ) ).

% sup_neg_inf
tff(fact_6488_sup__neg__inf,axiom,
    ! [P3: set_Pr8693737435421807431at_nat,Q3: set_Pr8693737435421807431at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(P3,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Q3),R2))
    <=> ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,P3),uminus4384627049435823934at_nat(Q3)),R2) ) ).

% sup_neg_inf
tff(fact_6489_sup__neg__inf,axiom,
    ! [P3: set_Pr4329608150637261639at_nat,Q3: set_Pr4329608150637261639at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(P3,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Q3),R2))
    <=> ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,P3),uminus935396558254630718at_nat(Q3)),R2) ) ).

% sup_neg_inf
tff(fact_6490_sup__neg__inf,axiom,
    ! [P3: set_int,Q3: set_int,R2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(P3),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Q3),R2)))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,P3),uminus1532241313380277803et_int(Q3))),R2)) ) ).

% sup_neg_inf
tff(fact_6491_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) = bot_bo2099793752762293965at_nat )
    <=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A3),uminus6524753893492686040at_nat(B2))) ) ).

% disjoint_eq_subset_Compl
tff(fact_6492_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ( inf_inf_set_real(A3,B2) = bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(A3),uminus612125837232591019t_real(B2))) ) ).

% disjoint_eq_subset_Compl
tff(fact_6493_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_bool,B2: set_bool] :
      ( ( inf_inf_set_bool(A3,B2) = bot_bot_set_bool )
    <=> pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),uminus5166143097222041326t_bool(B2))) ) ).

% disjoint_eq_subset_Compl
tff(fact_6494_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),uminus5710092332889474511et_nat(B2))) ) ).

% disjoint_eq_subset_Compl
tff(fact_6495_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(A3),uminus1532241313380277803et_int(B2))) ) ).

% disjoint_eq_subset_Compl
tff(fact_6496_card__Un__Int,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( finite8100373058378681591st_nat(B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(A3)),finite_card_list_nat(B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(sup_sup_set_list_nat(A3,B2))),finite_card_list_nat(inf_inf_set_list_nat(A3,B2))) ) ) ) ).

% card_Un_Int
tff(fact_6497_card__Un__Int,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_set_nat(A3)),finite_card_set_nat(B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_set_nat(sup_sup_set_set_nat(A3,B2))),finite_card_set_nat(inf_inf_set_set_nat(A3,B2))) ) ) ) ).

% card_Un_Int
tff(fact_6498_card__Un__Int,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_int(A3)),finite_card_int(B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2))),finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))) ) ) ) ).

% card_Un_Int
tff(fact_6499_card__Un__Int,axiom,
    ! [A3: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_complex(A3)),finite_card_complex(B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_complex(sup_sup_set_complex(A3,B2))),finite_card_complex(inf_inf_set_complex(A3,B2))) ) ) ) ).

% card_Un_Int
tff(fact_6500_card__Un__Int,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite121521170596916366d_enat(A3)),finite121521170596916366d_enat(B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite121521170596916366d_enat(sup_su4489774667511045786d_enat(A3,B2))),finite121521170596916366d_enat(inf_in8357106775501769908d_enat(A3,B2))) ) ) ) ).

% card_Un_Int
tff(fact_6501_card__Un__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B2))),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2))) ) ) ) ).

% card_Un_Int
tff(fact_6502_card__Un__Int,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2))),aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2))) ) ) ) ).

% card_Un_Int
tff(fact_6503_card__Un__Int,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( finite4392333629123659920at_nat(A3)
     => ( finite4392333629123659920at_nat(B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite1207074278014112911at_nat(A3)),finite1207074278014112911at_nat(B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite1207074278014112911at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2))),finite1207074278014112911at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B2))) ) ) ) ).

% card_Un_Int
tff(fact_6504_card__Un__Int,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( finite4343798906461161616at_nat(A3)
     => ( finite4343798906461161616at_nat(B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set168828743186997083at_nat(finite3771342082235030671at_nat,A3)),aa_set168828743186997083at_nat(finite3771342082235030671at_nat,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set168828743186997083at_nat(finite3771342082235030671at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2))),aa_set168828743186997083at_nat(finite3771342082235030671at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),B2))) ) ) ) ).

% card_Un_Int
tff(fact_6505_card__Diff__subset__Int,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] :
      ( finite8100373058378681591st_nat(inf_inf_set_list_nat(A3,B2))
     => ( finite_card_list_nat(minus_7954133019191499631st_nat(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A3)),finite_card_list_nat(inf_inf_set_list_nat(A3,B2))) ) ) ).

% card_Diff_subset_Int
tff(fact_6506_card__Diff__subset__Int,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,inf_inf_set_set_nat(A3,B2)))
     => ( finite_card_set_nat(minus_2163939370556025621et_nat(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A3)),finite_card_set_nat(inf_inf_set_set_nat(A3,B2))) ) ) ).

% card_Diff_subset_Int
tff(fact_6507_card__Diff__subset__Int,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)))
     => ( finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A3)),finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))) ) ) ).

% card_Diff_subset_Int
tff(fact_6508_card__Diff__subset__Int,axiom,
    ! [A3: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,inf_inf_set_complex(A3,B2)))
     => ( finite_card_complex(minus_811609699411566653omplex(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A3)),finite_card_complex(inf_inf_set_complex(A3,B2))) ) ) ).

% card_Diff_subset_Int
tff(fact_6509_card__Diff__subset__Int,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,inf_in8357106775501769908d_enat(A3,B2)))
     => ( finite121521170596916366d_enat(minus_925952699566721837d_enat(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite121521170596916366d_enat(A3)),finite121521170596916366d_enat(inf_in8357106775501769908d_enat(A3,B2))) ) ) ).

% card_Diff_subset_Int
tff(fact_6510_card__Diff__subset__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2)))
     => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2))) ) ) ).

% card_Diff_subset_Int
tff(fact_6511_card__Diff__subset__Int,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2)))
     => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2))) ) ) ).

% card_Diff_subset_Int
tff(fact_6512_card__Un__disjoint,axiom,
    ! [A3: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ( inf_inf_set_complex(A3,B2) = bot_bot_set_complex )
         => ( finite_card_complex(sup_sup_set_complex(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_complex(A3)),finite_card_complex(B2)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_6513_card__Un__disjoint,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( ( inf_in8357106775501769908d_enat(A3,B2) = bot_bo7653980558646680370d_enat )
         => ( finite121521170596916366d_enat(sup_su4489774667511045786d_enat(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite121521170596916366d_enat(A3)),finite121521170596916366d_enat(B2)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_6514_card__Un__disjoint,axiom,
    ! [A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(finite_finite_real,B2))
       => ( ( inf_inf_set_real(A3,B2) = bot_bot_set_real )
         => ( finite_card_real(sup_sup_set_real(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_real(A3)),finite_card_real(B2)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_6515_card__Un__disjoint,axiom,
    ! [A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( pp(aa_set_bool_bool(finite_finite_bool,B2))
       => ( ( inf_inf_set_bool(A3,B2) = bot_bot_set_bool )
         => ( finite_card_bool(sup_sup_set_bool(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_bool(A3)),finite_card_bool(B2)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_6516_card__Un__disjoint,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) = bot_bot_set_nat )
         => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(finite_card_nat,A3)),aa_set_nat_nat(finite_card_nat,B2)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_6517_card__Un__disjoint,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2) = bot_bot_set_int )
         => ( finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_int(A3)),finite_card_int(B2)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_6518_card__Un__disjoint,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( finite8100373058378681591st_nat(B2)
       => ( ( inf_inf_set_list_nat(A3,B2) = bot_bot_set_list_nat )
         => ( finite_card_list_nat(sup_sup_set_list_nat(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(A3)),finite_card_list_nat(B2)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_6519_card__Un__disjoint,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B2))
       => ( ( inf_inf_set_set_nat(A3,B2) = bot_bot_set_set_nat )
         => ( finite_card_set_nat(sup_sup_set_set_nat(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_set_nat(A3)),finite_card_set_nat(B2)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_6520_card__Un__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
       => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) = bot_bo2099793752762293965at_nat )
         => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B2)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_6521_card__Un__disjoint,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( finite4392333629123659920at_nat(A3)
     => ( finite4392333629123659920at_nat(B2)
       => ( ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B2) = bot_bo5327735625951526323at_nat )
         => ( finite1207074278014112911at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite1207074278014112911at_nat(A3)),finite1207074278014112911at_nat(B2)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_6522_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_6523_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X5),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [I2: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                   => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),X_1))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I2)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_12)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I2 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
                                & ! [X5: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N) = I2 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X5))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_6524_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X5),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [I2: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                   => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),X_1))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I2)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_12)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I2 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
                                & ! [X5: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N) = I2 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X5))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_6525_invar__vebt_Osimps,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(A12),A23))
    <=> ( ( ? [A6: bool,B6: bool] : A12 = vEBT_Leaf(A6,B6)
          & ( A23 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList4: list_VEBT_VEBT,N3: nat,Summary4: vEBT_VEBT] :
            ( ( A12 = vEBT_Node(none_P5556105721700978146at_nat,A23,TreeList4,Summary4) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList4)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X3),N3)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary4),N3))
            & ( size_s6755466524823107622T_VEBT(TreeList4) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3) )
            & ( A23 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
            & ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary4),X_1))
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList4)))
               => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) ) )
        | ? [TreeList4: list_VEBT_VEBT,N3: nat,Summary4: vEBT_VEBT] :
            ( ( A12 = vEBT_Node(none_P5556105721700978146at_nat,A23,TreeList4,Summary4) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList4)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X3),N3)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary4),aa_nat_nat(suc,N3)))
            & ( size_s6755466524823107622T_VEBT(TreeList4) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3)) )
            & ( A23 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) )
            & ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary4),X_1))
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList4)))
               => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) ) )
        | ? [TreeList4: list_VEBT_VEBT,N3: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A12 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A23,TreeList4,Summary4) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList4)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X3),N3)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary4),N3))
            & ( size_s6755466524823107622T_VEBT(TreeList4) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3) )
            & ( A23 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)))
               => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList4,I4)),X_1))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary4),I4)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList4)))
                 => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) ) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A23)))
            & ( ( Mi3 != Ma3 )
             => ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N3) = I4 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList4,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N3))) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N3) = I4 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList4,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N3))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X3))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma3)) ) ) ) ) ) )
        | ? [TreeList4: list_VEBT_VEBT,N3: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A12 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A23,TreeList4,Summary4) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList4)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X3),N3)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary4),aa_nat_nat(suc,N3)))
            & ( size_s6755466524823107622T_VEBT(TreeList4) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3)) )
            & ( A23 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) )
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3))))
               => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList4,I4)),X_1))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary4),I4)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList4)))
                 => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) ) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A23)))
            & ( ( Mi3 != Ma3 )
             => ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3))))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N3) = I4 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList4,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N3))) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N3) = I4 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList4,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N3))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X3))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma3)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_6526_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(A12),A23))
     => ( ( ? [A: bool,B: bool] : A12 = vEBT_Leaf(A,B)
         => ( A23 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
              ( ( A12 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2) )
             => ( ( A23 = Deg2 )
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                     => pp(aa_nat_bool(vEBT_invar_vebt(X4),N2)) )
                 => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M4))
                   => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4) )
                     => ( ( M4 = N2 )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M4) )
                         => ( ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_13))
                           => ~ ! [X4: vEBT_VEBT] :
                                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_13)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
                ( ( A12 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2) )
               => ( ( A23 = Deg2 )
                 => ( ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                       => pp(aa_nat_bool(vEBT_invar_vebt(X4),N2)) )
                   => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M4))
                     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4) )
                       => ( ( M4 = aa_nat_nat(suc,N2) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M4) )
                           => ( ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_13))
                             => ~ ! [X4: vEBT_VEBT] :
                                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                                   => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_13)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList,Summary2) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X4: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N2)) )
                     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M4))
                       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4) )
                         => ( ( M4 = N2 )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M4) )
                             => ( ! [I3: nat] :
                                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4)))
                                   => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),X_1))
                                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I3)) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X4: vEBT_VEBT] :
                                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                                       => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_13)) ) )
                                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I3: nat] :
                                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4)))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I3 )
                                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
                                                & ! [X4: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N2) = I3 )
                                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N2))) )
                                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X4))
                                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList,Summary2) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X4: vEBT_VEBT] :
                            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                           => pp(aa_nat_bool(vEBT_invar_vebt(X4),N2)) )
                       => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M4))
                         => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4) )
                           => ( ( M4 = aa_nat_nat(suc,N2) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M4) )
                               => ( ! [I3: nat] :
                                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4)))
                                     => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),X_1))
                                      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I3)) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X4: vEBT_VEBT] :
                                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                                         => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_13)) ) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I3: nat] :
                                                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4)))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I3 )
                                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
                                                  & ! [X4: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N2) = I3 )
                                                        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N2))) )
                                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X4))
                                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_6527_in__children__def,axiom,
    ! [N: nat,TreeList2: list_VEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(N,TreeList2),X))
    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N))) ) ).

% in_children_def
tff(fact_6528_del__x__mi__lets__in__not__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList2,H,Newnode) )
                     => ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_6529_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L) )
               => ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                 => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList2,H,Newnode) )
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                     => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_6530_set__bit__0,axiom,
    ! [A2: code_integer] : bit_se2793503036327961859nteger(zero_zero_nat,A2) = plus_p5714425477246183910nteger(one_one_Code_integer,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),divide6298287555418463151nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_6531_set__bit__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_6532_set__bit__0,axiom,
    ! [A2: int] : aa_int_int(bit_se7879613467334960850it_int(zero_zero_nat),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_6533_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) ).

% enat_ord_number(1)
tff(fact_6534_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se7879613467334960850it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% set_bit_nonnegative_int_iff
tff(fact_6535_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se7879613467334960850it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% set_bit_negative_int_iff
tff(fact_6536_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I))
     => ( list_u1324408373059187874T_VEBT(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_6537_list__update__beyond,axiom,
    ! [Xs: list_bool,I: nat,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Xs)),I))
     => ( list_update_bool(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_6538_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),I))
     => ( list_update_nat(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_6539_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) ).

% enat_ord_number(2)
tff(fact_6540_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_6541_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),I) = X ) ) ).

% nth_list_update_eq
tff(fact_6542_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I,X)),I))
      <=> pp(X) ) ) ).

% nth_list_update_eq
tff(fact_6543_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_6544_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_int(Xs)))
       => ( set_int2(list_update_int(list_update_int(Xs,I,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_6545_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs)))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,I))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_6546_set__swap,axiom,
    ! [I: nat,Xs: list_bool,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_bool(Xs)))
       => ( set_bool2(list_update_bool(list_update_bool(Xs,I,aa_nat_bool(nth_bool(Xs),J)),J,aa_nat_bool(nth_bool(Xs),I))) = set_bool2(Xs) ) ) ) ).

% set_swap
tff(fact_6547_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_nat(Xs)))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_6548_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,Z),Y))
     => ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(Y),Z)) = aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),Y)),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_6549_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,N),zero_z5237406670263579293d_enat))
    <=> ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_6550_i0__lb,axiom,
    ! [N: extended_enat] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),N)) ).

% i0_lb
tff(fact_6551_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_se7879613467334960850it_int(N),K))) ).

% set_bit_greater_eq
tff(fact_6552_list__update__code_I2_J,axiom,
    ! [X: int,Xs: list_int,Y: int] : list_update_int(cons_int(X,Xs),zero_zero_nat,Y) = cons_int(Y,Xs) ).

% list_update_code(2)
tff(fact_6553_list__update__code_I2_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat] : list_update_nat(cons_nat(X,Xs),zero_zero_nat,Y) = cons_nat(Y,Xs) ).

% list_update_code(2)
tff(fact_6554_list__update__code_I2_J,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: vEBT_VEBT] : list_u1324408373059187874T_VEBT(cons_VEBT_VEBT(X,Xs),zero_zero_nat,Y) = cons_VEBT_VEBT(Y,Xs) ).

% list_update_code(2)
tff(fact_6555_set__update__subsetI,axiom,
    ! [Xs: list_real,A3: set_real,X: real,I: nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Xs)),A3))
     => ( pp(aa_set_real_bool(member_real(X),A3))
       => pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X))),A3)) ) ) ).

% set_update_subsetI
tff(fact_6556_set__update__subsetI,axiom,
    ! [Xs: list_bool,A3: set_bool,X: bool,I: nat] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(set_bool2(Xs)),A3))
     => ( pp(aa_set_bool_bool(member_bool(X),A3))
       => pp(aa_set_bool_bool(ord_less_eq_set_bool(set_bool2(list_update_bool(Xs,I,X))),A3)) ) ) ).

% set_update_subsetI
tff(fact_6557_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A3: set_set_nat,X: set_nat,I: nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),A3))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X),A3))
       => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(list_update_set_nat(Xs,I,X))),A3)) ) ) ).

% set_update_subsetI
tff(fact_6558_set__update__subsetI,axiom,
    ! [Xs: list_nat,A3: set_nat,X: nat,I: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Xs)),A3))
     => ( pp(aa_set_nat_bool(member_nat(X),A3))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,I,X))),A3)) ) ) ).

% set_update_subsetI
tff(fact_6559_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A3: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),A3)
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A3))
       => ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X)),A3) ) ) ).

% set_update_subsetI
tff(fact_6560_set__update__subsetI,axiom,
    ! [Xs: list_int,A3: set_int,X: int,I: nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),A3))
     => ( pp(aa_set_int_bool(member_int(X),A3))
       => pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X))),A3)) ) ) ).

% set_update_subsetI
tff(fact_6561_set__update__subset__insert,axiom,
    ! [Xs: list_P6011104703257516679at_nat,I: nat,X: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,I,X))),insert8211810215607154385at_nat(X,set_Pr5648618587558075414at_nat(Xs)))) ).

% set_update_subset_insert
tff(fact_6562_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,X: real] : pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X))),insert_real(X,set_real2(Xs)))) ).

% set_update_subset_insert
tff(fact_6563_set__update__subset__insert,axiom,
    ! [Xs: list_bool,I: nat,X: bool] : pp(aa_set_bool_bool(ord_less_eq_set_bool(set_bool2(list_update_bool(Xs,I,X))),insert_bool(X,set_bool2(Xs)))) ).

% set_update_subset_insert
tff(fact_6564_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,I,X))),insert_nat(X,set_nat2(Xs)))) ).

% set_update_subset_insert
tff(fact_6565_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X)),insert_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))) ).

% set_update_subset_insert
tff(fact_6566_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X: int] : pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X))),insert_int(X,set_int2(Xs)))) ).

% set_update_subset_insert
tff(fact_6567_set__update__memI,axiom,
    ! [N: nat,Xs: list_real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => pp(aa_set_real_bool(member_real(X),set_real2(list_update_real(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_6568_set__update__memI,axiom,
    ! [N: nat,Xs: list_set_nat,X: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
     => pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(list_update_set_nat(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_6569_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => pp(aa_set_int_bool(member_int(X),set_int2(list_update_int(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_6570_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_6571_set__update__memI,axiom,
    ! [N: nat,Xs: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => pp(aa_set_bool_bool(member_bool(X),set_bool2(list_update_bool(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_6572_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => pp(aa_set_nat_bool(member_nat(X),set_nat2(list_update_nat(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_6573_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,J: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( ( ( I = J )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J) = X ) )
        & ( ( I != J )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J) = aa_nat_int(nth_int(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_6574_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( ( I = J )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J) = X ) )
        & ( ( I != J )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J) = nth_VEBT_VEBT(Xs,J) ) ) ) ) ).

% nth_list_update
tff(fact_6575_nth__list__update,axiom,
    ! [I: nat,Xs: list_bool,X: bool,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I,X)),J))
      <=> ( ( ( I = J )
           => pp(X) )
          & ( ( I != J )
           => pp(aa_nat_bool(nth_bool(Xs),J)) ) ) ) ) ).

% nth_list_update
tff(fact_6576_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( ( ( I = J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = X ) )
        & ( ( I != J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_6577_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( ( list_update_int(Xs,I,X) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_6578_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I,X) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I) = X ) ) ) ).

% list_update_same_conv
tff(fact_6579_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( ( list_update_bool(Xs,I,X) = Xs )
      <=> ( pp(aa_nat_bool(nth_bool(Xs),I))
        <=> pp(X) ) ) ) ).

% list_update_same_conv
tff(fact_6580_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( ( list_update_nat(Xs,I,X) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_6581_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_6582_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mi),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_6583_sup__bot__left,axiom,
    ! [X: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,bot_bo5327735625951526323at_nat),X) = X ).

% sup_bot_left
tff(fact_6584_sup__bot__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),X) = X ).

% sup_bot_left
tff(fact_6585_sup__bot__left,axiom,
    ! [X: set_real] : sup_sup_set_real(bot_bot_set_real,X) = X ).

% sup_bot_left
tff(fact_6586_sup__bot__left,axiom,
    ! [X: set_bool] : sup_sup_set_bool(bot_bot_set_bool,X) = X ).

% sup_bot_left
tff(fact_6587_sup__bot__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),X) = X ).

% sup_bot_left
tff(fact_6588_sup__bot__left,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),X) = X ).

% sup_bot_left
tff(fact_6589_sup__bot__right,axiom,
    ! [X: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),bot_bo5327735625951526323at_nat) = X ).

% sup_bot_right
tff(fact_6590_sup__bot__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),bot_bo228742789529271731at_nat) = X ).

% sup_bot_right
tff(fact_6591_sup__bot__right,axiom,
    ! [X: set_real] : sup_sup_set_real(X,bot_bot_set_real) = X ).

% sup_bot_right
tff(fact_6592_sup__bot__right,axiom,
    ! [X: set_bool] : sup_sup_set_bool(X,bot_bot_set_bool) = X ).

% sup_bot_right
tff(fact_6593_sup__bot__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),bot_bot_set_nat) = X ).

% sup_bot_right
tff(fact_6594_sup__bot__right,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),bot_bot_set_int) = X ).

% sup_bot_right
tff(fact_6595_bot__eq__sup__iff,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ( bot_bo5327735625951526323at_nat = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y) )
    <=> ( ( X = bot_bo5327735625951526323at_nat )
        & ( Y = bot_bo5327735625951526323at_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_6596_bot__eq__sup__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) )
    <=> ( ( X = bot_bo228742789529271731at_nat )
        & ( Y = bot_bo228742789529271731at_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_6597_bot__eq__sup__iff,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( bot_bot_set_real = sup_sup_set_real(X,Y) )
    <=> ( ( X = bot_bot_set_real )
        & ( Y = bot_bot_set_real ) ) ) ).

% bot_eq_sup_iff
tff(fact_6598_bot__eq__sup__iff,axiom,
    ! [X: set_bool,Y: set_bool] :
      ( ( bot_bot_set_bool = sup_sup_set_bool(X,Y) )
    <=> ( ( X = bot_bot_set_bool )
        & ( Y = bot_bot_set_bool ) ) ) ).

% bot_eq_sup_iff
tff(fact_6599_bot__eq__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) )
    <=> ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_6600_bot__eq__sup__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) )
    <=> ( ( X = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% bot_eq_sup_iff
tff(fact_6601_sup__eq__bot__iff,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y) = bot_bo5327735625951526323at_nat )
    <=> ( ( X = bot_bo5327735625951526323at_nat )
        & ( Y = bot_bo5327735625951526323at_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_6602_sup__eq__bot__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = bot_bo228742789529271731at_nat )
    <=> ( ( X = bot_bo228742789529271731at_nat )
        & ( Y = bot_bo228742789529271731at_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_6603_sup__eq__bot__iff,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( sup_sup_set_real(X,Y) = bot_bot_set_real )
    <=> ( ( X = bot_bot_set_real )
        & ( Y = bot_bot_set_real ) ) ) ).

% sup_eq_bot_iff
tff(fact_6604_sup__eq__bot__iff,axiom,
    ! [X: set_bool,Y: set_bool] :
      ( ( sup_sup_set_bool(X,Y) = bot_bot_set_bool )
    <=> ( ( X = bot_bot_set_bool )
        & ( Y = bot_bot_set_bool ) ) ) ).

% sup_eq_bot_iff
tff(fact_6605_sup__eq__bot__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = bot_bot_set_nat )
    <=> ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_6606_sup__eq__bot__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = bot_bot_set_int )
    <=> ( ( X = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% sup_eq_bot_iff
tff(fact_6607_le__inf__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Z)) ) ) ).

% le_inf_iff
tff(fact_6608_le__inf__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)))
    <=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),Y))
        & pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),Z)) ) ) ).

% le_inf_iff
tff(fact_6609_le__inf__iff,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Y),Z)))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
        & pp(aa_set_int_bool(ord_less_eq_set_int(X),Z)) ) ) ).

% le_inf_iff
tff(fact_6610_le__inf__iff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z)) ) ) ).

% le_inf_iff
tff(fact_6611_le__inf__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z)) ) ) ).

% le_inf_iff
tff(fact_6612_le__inf__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z)) ) ) ).

% le_inf_iff
tff(fact_6613_inf_Obounded__iff,axiom,
    ! [A2: set_nat,B3: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),C)))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),C)) ) ) ).

% inf.bounded_iff
tff(fact_6614_inf_Obounded__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),C)))
    <=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B3))
        & pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),C)) ) ) ).

% inf.bounded_iff
tff(fact_6615_inf_Obounded__iff,axiom,
    ! [A2: set_int,B3: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B3),C)))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
        & pp(aa_set_int_bool(ord_less_eq_set_int(A2),C)) ) ) ).

% inf.bounded_iff
tff(fact_6616_inf_Obounded__iff,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B3),C)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),C)) ) ) ).

% inf.bounded_iff
tff(fact_6617_inf_Obounded__iff,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B3),C)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C)) ) ) ).

% inf.bounded_iff
tff(fact_6618_inf_Obounded__iff,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(inf_inf_int,B3),C)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C)) ) ) ).

% inf.bounded_iff
tff(fact_6619_le__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Z))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),Z)) ) ) ).

% le_sup_iff
tff(fact_6620_le__sup__iff,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y),Z)
    <=> ( ord_le3000389064537975527at_nat(X,Z)
        & ord_le3000389064537975527at_nat(Y,Z) ) ) ).

% le_sup_iff
tff(fact_6621_le__sup__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y),Z)
    <=> ( ord_le1268244103169919719at_nat(X,Z)
        & ord_le1268244103169919719at_nat(Y,Z) ) ) ).

% le_sup_iff
tff(fact_6622_le__sup__iff,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y)),Z))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Z))
        & pp(aa_set_int_bool(ord_less_eq_set_int(Y),Z)) ) ) ).

% le_sup_iff
tff(fact_6623_le__sup__iff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)),Z))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ) ).

% le_sup_iff
tff(fact_6624_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),Z))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z)) ) ) ).

% le_sup_iff
tff(fact_6625_le__sup__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),Z))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z)) ) ) ).

% le_sup_iff
tff(fact_6626_sup_Obounded__iff,axiom,
    ! [B3: set_nat,C: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),C)),A2))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),A2))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),A2)) ) ) ).

% sup.bounded_iff
tff(fact_6627_sup_Obounded__iff,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B3),C),A2)
    <=> ( ord_le3000389064537975527at_nat(B3,A2)
        & ord_le3000389064537975527at_nat(C,A2) ) ) ).

% sup.bounded_iff
tff(fact_6628_sup_Obounded__iff,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B3),C),A2)
    <=> ( ord_le1268244103169919719at_nat(B3,A2)
        & ord_le1268244103169919719at_nat(C,A2) ) ) ).

% sup.bounded_iff
tff(fact_6629_sup_Obounded__iff,axiom,
    ! [B3: set_int,C: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B3),C)),A2))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
        & pp(aa_set_int_bool(ord_less_eq_set_int(C),A2)) ) ) ).

% sup.bounded_iff
tff(fact_6630_sup_Obounded__iff,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B3),C)),A2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A2)) ) ) ).

% sup.bounded_iff
tff(fact_6631_sup_Obounded__iff,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B3),C)),A2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2)) ) ) ).

% sup.bounded_iff
tff(fact_6632_sup_Obounded__iff,axiom,
    ! [B3: int,C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B3),C)),A2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2)) ) ) ).

% sup.bounded_iff
tff(fact_6633_inf__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),X) = bot_bo2099793752762293965at_nat ).

% inf_bot_left
tff(fact_6634_inf__bot__left,axiom,
    ! [X: set_real] : inf_inf_set_real(bot_bot_set_real,X) = bot_bot_set_real ).

% inf_bot_left
tff(fact_6635_inf__bot__left,axiom,
    ! [X: set_bool] : inf_inf_set_bool(bot_bot_set_bool,X) = bot_bot_set_bool ).

% inf_bot_left
tff(fact_6636_inf__bot__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),X) = bot_bot_set_nat ).

% inf_bot_left
tff(fact_6637_inf__bot__left,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,bot_bot_set_int),X) = bot_bot_set_int ).

% inf_bot_left
tff(fact_6638_inf__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% inf_bot_right
tff(fact_6639_inf__bot__right,axiom,
    ! [X: set_real] : inf_inf_set_real(X,bot_bot_set_real) = bot_bot_set_real ).

% inf_bot_right
tff(fact_6640_inf__bot__right,axiom,
    ! [X: set_bool] : inf_inf_set_bool(X,bot_bot_set_bool) = bot_bot_set_bool ).

% inf_bot_right
tff(fact_6641_inf__bot__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),bot_bot_set_nat) = bot_bot_set_nat ).

% inf_bot_right
tff(fact_6642_inf__bot__right,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),bot_bot_set_int) = bot_bot_set_int ).

% inf_bot_right
tff(fact_6643_sup__bot_Oright__neutral,axiom,
    ! [A2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),bot_bo5327735625951526323at_nat) = A2 ).

% sup_bot.right_neutral
tff(fact_6644_sup__bot_Oright__neutral,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),bot_bo228742789529271731at_nat) = A2 ).

% sup_bot.right_neutral
tff(fact_6645_sup__bot_Oright__neutral,axiom,
    ! [A2: set_real] : sup_sup_set_real(A2,bot_bot_set_real) = A2 ).

% sup_bot.right_neutral
tff(fact_6646_sup__bot_Oright__neutral,axiom,
    ! [A2: set_bool] : sup_sup_set_bool(A2,bot_bot_set_bool) = A2 ).

% sup_bot.right_neutral
tff(fact_6647_sup__bot_Oright__neutral,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),bot_bot_set_nat) = A2 ).

% sup_bot.right_neutral
tff(fact_6648_sup__bot_Oright__neutral,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),bot_bot_set_int) = A2 ).

% sup_bot.right_neutral
tff(fact_6649_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( bot_bo5327735625951526323at_nat = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3) )
    <=> ( ( A2 = bot_bo5327735625951526323at_nat )
        & ( B3 = bot_bo5327735625951526323at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_6650_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3) )
    <=> ( ( A2 = bot_bo228742789529271731at_nat )
        & ( B3 = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_6651_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ( bot_bot_set_real = sup_sup_set_real(A2,B3) )
    <=> ( ( A2 = bot_bot_set_real )
        & ( B3 = bot_bot_set_real ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_6652_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_bool,B3: set_bool] :
      ( ( bot_bot_set_bool = sup_sup_set_bool(A2,B3) )
    <=> ( ( A2 = bot_bot_set_bool )
        & ( B3 = bot_bot_set_bool ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_6653_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) )
    <=> ( ( A2 = bot_bot_set_nat )
        & ( B3 = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_6654_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3) )
    <=> ( ( A2 = bot_bot_set_int )
        & ( B3 = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_6655_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,bot_bo5327735625951526323at_nat),A2) = A2 ).

% sup_bot.left_neutral
tff(fact_6656_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),A2) = A2 ).

% sup_bot.left_neutral
tff(fact_6657_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_real] : sup_sup_set_real(bot_bot_set_real,A2) = A2 ).

% sup_bot.left_neutral
tff(fact_6658_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_bool] : sup_sup_set_bool(bot_bot_set_bool,A2) = A2 ).

% sup_bot.left_neutral
tff(fact_6659_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),A2) = A2 ).

% sup_bot.left_neutral
tff(fact_6660_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),A2) = A2 ).

% sup_bot.left_neutral
tff(fact_6661_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3) = bot_bo5327735625951526323at_nat )
    <=> ( ( A2 = bot_bo5327735625951526323at_nat )
        & ( B3 = bot_bo5327735625951526323at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_6662_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3) = bot_bo228742789529271731at_nat )
    <=> ( ( A2 = bot_bo228742789529271731at_nat )
        & ( B3 = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_6663_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ( sup_sup_set_real(A2,B3) = bot_bot_set_real )
    <=> ( ( A2 = bot_bot_set_real )
        & ( B3 = bot_bot_set_real ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_6664_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_bool,B3: set_bool] :
      ( ( sup_sup_set_bool(A2,B3) = bot_bot_set_bool )
    <=> ( ( A2 = bot_bot_set_bool )
        & ( B3 = bot_bot_set_bool ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_6665_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) = bot_bot_set_nat )
    <=> ( ( A2 = bot_bot_set_nat )
        & ( B3 = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_6666_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3) = bot_bot_set_int )
    <=> ( ( A2 = bot_bot_set_int )
        & ( B3 = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_6667_max_Obounded__iff,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B3),C)),A2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A2)) ) ) ).

% max.bounded_iff
tff(fact_6668_max_Obounded__iff,axiom,
    ! [B3: num,C: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B3),C)),A2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A2)) ) ) ).

% max.bounded_iff
tff(fact_6669_max_Obounded__iff,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B3),C)),A2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2)) ) ) ).

% max.bounded_iff
tff(fact_6670_max_Obounded__iff,axiom,
    ! [B3: int,C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B3),C)),A2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2)) ) ) ).

% max.bounded_iff
tff(fact_6671_max_Oabsorb2,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3) = B3 ) ) ).

% max.absorb2
tff(fact_6672_max_Oabsorb2,axiom,
    ! [A2: num,B3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3) = B3 ) ) ).

% max.absorb2
tff(fact_6673_max_Oabsorb2,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3) = B3 ) ) ).

% max.absorb2
tff(fact_6674_max_Oabsorb2,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3) = B3 ) ) ).

% max.absorb2
tff(fact_6675_max_Oabsorb1,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3) = A2 ) ) ).

% max.absorb1
tff(fact_6676_max_Oabsorb1,axiom,
    ! [B3: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3) = A2 ) ) ).

% max.absorb1
tff(fact_6677_max_Oabsorb1,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3) = A2 ) ) ).

% max.absorb1
tff(fact_6678_max_Oabsorb1,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3) = A2 ) ) ).

% max.absorb1
tff(fact_6679_max__less__iff__conj,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)),Z))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_6680_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y)),Z))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_6681_max__less__iff__conj,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y)),Z))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_6682_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)),Z))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_6683_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_6684_max_Oabsorb4,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A2),B3) = B3 ) ) ).

% max.absorb4
tff(fact_6685_max_Oabsorb4,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3) = B3 ) ) ).

% max.absorb4
tff(fact_6686_max_Oabsorb4,axiom,
    ! [A2: num,B3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B3))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3) = B3 ) ) ).

% max.absorb4
tff(fact_6687_max_Oabsorb4,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3) = B3 ) ) ).

% max.absorb4
tff(fact_6688_max_Oabsorb4,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3) = B3 ) ) ).

% max.absorb4
tff(fact_6689_max_Oabsorb3,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A2),B3) = A2 ) ) ).

% max.absorb3
tff(fact_6690_max_Oabsorb3,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3) = A2 ) ) ).

% max.absorb3
tff(fact_6691_max_Oabsorb3,axiom,
    ! [B3: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),A2))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3) = A2 ) ) ).

% max.absorb3
tff(fact_6692_max_Oabsorb3,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3) = A2 ) ) ).

% max.absorb3
tff(fact_6693_max_Oabsorb3,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3) = A2 ) ) ).

% max.absorb3
tff(fact_6694_max__bot,axiom,
    ! [X: set_real] : ord_max_set_real(bot_bot_set_real,X) = X ).

% max_bot
tff(fact_6695_max__bot,axiom,
    ! [X: set_bool] : ord_max_set_bool(bot_bot_set_bool,X) = X ).

% max_bot
tff(fact_6696_max__bot,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,bot_bot_set_nat),X) = X ).

% max_bot
tff(fact_6697_max__bot,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,bot_bot_set_int),X) = X ).

% max_bot
tff(fact_6698_max__bot,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),X) = X ).

% max_bot
tff(fact_6699_max__bot2,axiom,
    ! [X: set_real] : ord_max_set_real(X,bot_bot_set_real) = X ).

% max_bot2
tff(fact_6700_max__bot2,axiom,
    ! [X: set_bool] : ord_max_set_bool(X,bot_bot_set_bool) = X ).

% max_bot2
tff(fact_6701_max__bot2,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,X),bot_bot_set_nat) = X ).

% max_bot2
tff(fact_6702_max__bot2,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,X),bot_bot_set_int) = X ).

% max_bot2
tff(fact_6703_max__bot2,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),bot_bot_nat) = X ).

% max_bot2
tff(fact_6704_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)) ).

% max_Suc_Suc
tff(fact_6705_max__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3) = zero_zero_nat )
    <=> ( ( A2 = zero_zero_nat )
        & ( B3 = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_6706_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),A2) = A2 ).

% max_nat.left_neutral
tff(fact_6707_max__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B3: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3) )
    <=> ( ( A2 = zero_zero_nat )
        & ( B3 = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_6708_max__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),zero_zero_nat) = A2 ).

% max_nat.right_neutral
tff(fact_6709_max__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N) = N ).

% max_0L
tff(fact_6710_max__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),zero_zero_nat) = N ).

% max_0R
tff(fact_6711_i0__less,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),N))
    <=> ( N != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_6712_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)) = aa_num_real(numeral_numeral_real,V) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)) = aa_num_real(numeral_numeral_real,U) ) ) ) ).

% max_number_of(1)
tff(fact_6713_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)))
       => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)) = numera1916890842035813515d_enat(V) ) )
      & ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)))
       => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)) = numera1916890842035813515d_enat(U) ) ) ) ).

% max_number_of(1)
tff(fact_6714_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),numera6620942414471956472nteger(V)) = numera6620942414471956472nteger(V) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),numera6620942414471956472nteger(V)) = numera6620942414471956472nteger(U) ) ) ) ).

% max_number_of(1)
tff(fact_6715_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = aa_num_rat(numeral_numeral_rat,V) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = aa_num_rat(numeral_numeral_rat,U) ) ) ) ).

% max_number_of(1)
tff(fact_6716_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = aa_num_nat(numeral_numeral_nat,V) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = aa_num_nat(numeral_numeral_nat,U) ) ) ) ).

% max_number_of(1)
tff(fact_6717_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,V) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,U) ) ) ) ).

% max_number_of(1)
tff(fact_6718_max__0__1_I4_J,axiom,
    ! [X: num] : aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_num_rat(numeral_numeral_rat,X)),zero_zero_rat) = aa_num_rat(numeral_numeral_rat,X) ).

% max_0_1(4)
tff(fact_6719_max__0__1_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(4)
tff(fact_6720_max__0__1_I4_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,X)),zero_zero_real) = aa_num_real(numeral_numeral_real,X) ).

% max_0_1(4)
tff(fact_6721_max__0__1_I4_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,X)),zero_zero_int) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(4)
tff(fact_6722_max__0__1_I4_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(X)),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(X) ).

% max_0_1(4)
tff(fact_6723_max__0__1_I4_J,axiom,
    ! [X: num] : ord_max_Code_integer(numera6620942414471956472nteger(X),zero_z3403309356797280102nteger) = numera6620942414471956472nteger(X) ).

% max_0_1(4)
tff(fact_6724_max__0__1_I3_J,axiom,
    ! [X: num] : aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,X)) = aa_num_rat(numeral_numeral_rat,X) ).

% max_0_1(3)
tff(fact_6725_max__0__1_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(3)
tff(fact_6726_max__0__1_I3_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),aa_num_real(numeral_numeral_real,X)) = aa_num_real(numeral_numeral_real,X) ).

% max_0_1(3)
tff(fact_6727_max__0__1_I3_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),aa_num_int(numeral_numeral_int,X)) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(3)
tff(fact_6728_max__0__1_I3_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).

% max_0_1(3)
tff(fact_6729_max__0__1_I3_J,axiom,
    ! [X: num] : ord_max_Code_integer(zero_z3403309356797280102nteger,numera6620942414471956472nteger(X)) = numera6620942414471956472nteger(X) ).

% max_0_1(3)
tff(fact_6730_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_6731_max__0__1_I2_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,one_one_rat),zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_6732_max__0__1_I2_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_6733_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_6734_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_6735_max__0__1_I1_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,zero_zero_rat),one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_6736_max__0__1_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_6737_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_6738_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_num_real(numeral_numeral_real,U) ) ) ) ).

% max_number_of(2)
tff(fact_6739_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = numera6620942414471956472nteger(U) ) ) ) ).

% max_number_of(2)
tff(fact_6740_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_num_rat(numeral_numeral_rat,U) ) ) ) ).

% max_number_of(2)
tff(fact_6741_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_num_int(numeral_numeral_int,U) ) ) ) ).

% max_number_of(2)
tff(fact_6742_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)) = aa_num_real(numeral_numeral_real,V) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U)) ) ) ) ).

% max_number_of(3)
tff(fact_6743_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = numera6620942414471956472nteger(V) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)) ) ) ) ).

% max_number_of(3)
tff(fact_6744_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)) = aa_num_rat(numeral_numeral_rat,V) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U)) ) ) ) ).

% max_number_of(3)
tff(fact_6745_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,V) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U)) ) ) ) ).

% max_number_of(3)
tff(fact_6746_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U)) ) ) ) ).

% max_number_of(4)
tff(fact_6747_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)) ) ) ) ).

% max_number_of(4)
tff(fact_6748_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U)) ) ) ) ).

% max_number_of(4)
tff(fact_6749_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U)) ) ) ) ).

% max_number_of(4)
tff(fact_6750_max_OcoboundedI2,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3))) ) ).

% max.coboundedI2
tff(fact_6751_max_OcoboundedI2,axiom,
    ! [C: num,B3: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3))) ) ).

% max.coboundedI2
tff(fact_6752_max_OcoboundedI2,axiom,
    ! [C: nat,B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3))) ) ).

% max.coboundedI2
tff(fact_6753_max_OcoboundedI2,axiom,
    ! [C: int,B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3))) ) ).

% max.coboundedI2
tff(fact_6754_max_OcoboundedI1,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3))) ) ).

% max.coboundedI1
tff(fact_6755_max_OcoboundedI1,axiom,
    ! [C: num,A2: num,B3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3))) ) ).

% max.coboundedI1
tff(fact_6756_max_OcoboundedI1,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3))) ) ).

% max.coboundedI1
tff(fact_6757_max_OcoboundedI1,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3))) ) ).

% max.coboundedI1
tff(fact_6758_max_Oabsorb__iff2,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3) = B3 ) ) ).

% max.absorb_iff2
tff(fact_6759_max_Oabsorb__iff2,axiom,
    ! [A2: num,B3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3) = B3 ) ) ).

% max.absorb_iff2
tff(fact_6760_max_Oabsorb__iff2,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3) = B3 ) ) ).

% max.absorb_iff2
tff(fact_6761_max_Oabsorb__iff2,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3) = B3 ) ) ).

% max.absorb_iff2
tff(fact_6762_max_Oabsorb__iff1,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3) = A2 ) ) ).

% max.absorb_iff1
tff(fact_6763_max_Oabsorb__iff1,axiom,
    ! [B3: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2))
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3) = A2 ) ) ).

% max.absorb_iff1
tff(fact_6764_max_Oabsorb__iff1,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3) = A2 ) ) ).

% max.absorb_iff1
tff(fact_6765_max_Oabsorb__iff1,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3) = A2 ) ) ).

% max.absorb_iff1
tff(fact_6766_le__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_6767_le__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_6768_le__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_6769_le__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_6770_max_Ocobounded2,axiom,
    ! [B3: rat,A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3))) ).

% max.cobounded2
tff(fact_6771_max_Ocobounded2,axiom,
    ! [B3: num,A2: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3))) ).

% max.cobounded2
tff(fact_6772_max_Ocobounded2,axiom,
    ! [B3: nat,A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3))) ).

% max.cobounded2
tff(fact_6773_max_Ocobounded2,axiom,
    ! [B3: int,A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3))) ).

% max.cobounded2
tff(fact_6774_max_Ocobounded1,axiom,
    ! [A2: rat,B3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3))) ).

% max.cobounded1
tff(fact_6775_max_Ocobounded1,axiom,
    ! [A2: num,B3: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3))) ).

% max.cobounded1
tff(fact_6776_max_Ocobounded1,axiom,
    ! [A2: nat,B3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3))) ).

% max.cobounded1
tff(fact_6777_max_Ocobounded1,axiom,
    ! [A2: int,B3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3))) ).

% max.cobounded1
tff(fact_6778_max_Oorder__iff,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
    <=> ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3) ) ) ).

% max.order_iff
tff(fact_6779_max_Oorder__iff,axiom,
    ! [B3: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2))
    <=> ( A2 = aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3) ) ) ).

% max.order_iff
tff(fact_6780_max_Oorder__iff,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
    <=> ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3) ) ) ).

% max.order_iff
tff(fact_6781_max_Oorder__iff,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
    <=> ( A2 = aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3) ) ) ).

% max.order_iff
tff(fact_6782_max_OboundedI,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B3),C)),A2)) ) ) ).

% max.boundedI
tff(fact_6783_max_OboundedI,axiom,
    ! [B3: num,A2: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B3),C)),A2)) ) ) ).

% max.boundedI
tff(fact_6784_max_OboundedI,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B3),C)),A2)) ) ) ).

% max.boundedI
tff(fact_6785_max_OboundedI,axiom,
    ! [B3: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B3),C)),A2)) ) ) ).

% max.boundedI
tff(fact_6786_max_OboundedE,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B3),C)),A2))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A2)) ) ) ).

% max.boundedE
tff(fact_6787_max_OboundedE,axiom,
    ! [B3: num,C: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B3),C)),A2))
     => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A2)) ) ) ).

% max.boundedE
tff(fact_6788_max_OboundedE,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B3),C)),A2))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2)) ) ) ).

% max.boundedE
tff(fact_6789_max_OboundedE,axiom,
    ! [B3: int,C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B3),C)),A2))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2)) ) ) ).

% max.boundedE
tff(fact_6790_max_OorderI,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2)) ) ).

% max.orderI
tff(fact_6791_max_OorderI,axiom,
    ! [A2: num,B3: num] :
      ( ( A2 = aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3) )
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2)) ) ).

% max.orderI
tff(fact_6792_max_OorderI,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2)) ) ).

% max.orderI
tff(fact_6793_max_OorderI,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2)) ) ).

% max.orderI
tff(fact_6794_max_OorderE,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
     => ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3) ) ) ).

% max.orderE
tff(fact_6795_max_OorderE,axiom,
    ! [B3: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A2))
     => ( A2 = aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3) ) ) ).

% max.orderE
tff(fact_6796_max_OorderE,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
     => ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3) ) ) ).

% max.orderE
tff(fact_6797_max_OorderE,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
     => ( A2 = aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3) ) ) ).

% max.orderE
tff(fact_6798_max_Omono,axiom,
    ! [C: rat,A2: rat,D: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,C),D)),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3))) ) ) ).

% max.mono
tff(fact_6799_max_Omono,axiom,
    ! [C: num,A2: num,D: num,B3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,D),B3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,C),D)),aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3))) ) ) ).

% max.mono
tff(fact_6800_max_Omono,axiom,
    ! [C: nat,A2: nat,D: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3))) ) ) ).

% max.mono
tff(fact_6801_max_Omono,axiom,
    ! [C: int,A2: int,D: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C),D)),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3))) ) ) ).

% max.mono
tff(fact_6802_not__iless0,axiom,
    ! [N: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,N),zero_z5237406670263579293d_enat)) ).

% not_iless0
tff(fact_6803_enat__less__induct,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,N: extended_enat] :
      ( ! [N2: extended_enat] :
          ( ! [M2: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,M2),N2))
             => pp(aa_Ext2772979874139974381t_bool(P,M2)) )
         => pp(aa_Ext2772979874139974381t_bool(P,N2)) )
     => pp(aa_Ext2772979874139974381t_bool(P,N)) ) ).

% enat_less_induct
tff(fact_6804_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(M),N)))
    <=> ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),M))
        & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),N)) ) ) ).

% enat_0_less_mult_iff
tff(fact_6805_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_6806_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).

% of_nat_max
tff(fact_6807_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).

% of_nat_max
tff(fact_6808_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).

% of_nat_max
tff(fact_6809_less__max__iff__disj,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_6810_less__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_6811_less__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_6812_less__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_6813_less__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_6814_max_Ostrict__boundedE,axiom,
    ! [B3: real,C: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B3),C)),A2))
     => ~ ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
         => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A2)) ) ) ).

% max.strict_boundedE
tff(fact_6815_max_Ostrict__boundedE,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B3),C)),A2))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A2)) ) ) ).

% max.strict_boundedE
tff(fact_6816_max_Ostrict__boundedE,axiom,
    ! [B3: num,C: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B3),C)),A2))
     => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),A2))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A2)) ) ) ).

% max.strict_boundedE
tff(fact_6817_max_Ostrict__boundedE,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B3),C)),A2))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A2)) ) ) ).

% max.strict_boundedE
tff(fact_6818_max_Ostrict__boundedE,axiom,
    ! [B3: int,C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B3),C)),A2))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A2)) ) ) ).

% max.strict_boundedE
tff(fact_6819_max_Ostrict__order__iff,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
    <=> ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% max.strict_order_iff
tff(fact_6820_max_Ostrict__order__iff,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
    <=> ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% max.strict_order_iff
tff(fact_6821_max_Ostrict__order__iff,axiom,
    ! [B3: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),A2))
    <=> ( ( A2 = aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% max.strict_order_iff
tff(fact_6822_max_Ostrict__order__iff,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
    <=> ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% max.strict_order_iff
tff(fact_6823_max_Ostrict__order__iff,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
    <=> ( ( A2 = aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% max.strict_order_iff
tff(fact_6824_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A2),B3))) ) ).

% max.strict_coboundedI1
tff(fact_6825_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3))) ) ).

% max.strict_coboundedI1
tff(fact_6826_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A2: num,B3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3))) ) ).

% max.strict_coboundedI1
tff(fact_6827_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3))) ) ).

% max.strict_coboundedI1
tff(fact_6828_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3))) ) ).

% max.strict_coboundedI1
tff(fact_6829_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A2),B3))) ) ).

% max.strict_coboundedI2
tff(fact_6830_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3))) ) ).

% max.strict_coboundedI2
tff(fact_6831_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B3: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3))) ) ).

% max.strict_coboundedI2
tff(fact_6832_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3))) ) ).

% max.strict_coboundedI2
tff(fact_6833_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3))) ) ).

% max.strict_coboundedI2
tff(fact_6834_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_6835_max__def,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,A2),B3) = B3 ) )
      & ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,A2),B3) = A2 ) ) ) ).

% max_def
tff(fact_6836_max__def,axiom,
    ! [A2: rat,B3: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3) = B3 ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A2),B3) = A2 ) ) ) ).

% max_def
tff(fact_6837_max__def,axiom,
    ! [A2: num,B3: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3) = B3 ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B3))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B3) = A2 ) ) ) ).

% max_def
tff(fact_6838_max__def,axiom,
    ! [A2: nat,B3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3) = B3 ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B3) = A2 ) ) ) ).

% max_def
tff(fact_6839_max__def,axiom,
    ! [A2: int,B3: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3) = B3 ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B3) = A2 ) ) ) ).

% max_def
tff(fact_6840_max__absorb1,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,X),Y) = X ) ) ).

% max_absorb1
tff(fact_6841_max__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y) = X ) ) ).

% max_absorb1
tff(fact_6842_max__absorb1,axiom,
    ! [Y: num,X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y) = X ) ) ).

% max_absorb1
tff(fact_6843_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y) = X ) ) ).

% max_absorb1
tff(fact_6844_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y) = X ) ) ).

% max_absorb1
tff(fact_6845_max__absorb2,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_6846_max__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_6847_max__absorb2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_6848_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_6849_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_6850_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),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,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Z)) ).

% max_add_distrib_right
tff(fact_6851_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,Y),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Z)) ).

% max_add_distrib_right
tff(fact_6852_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(ord_max_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)) ).

% max_add_distrib_right
tff(fact_6853_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)) ).

% max_add_distrib_right
tff(fact_6854_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),Z)) ).

% max_add_distrib_left
tff(fact_6855_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y)),Z) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),Z)) ).

% max_add_distrib_left
tff(fact_6856_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ).

% max_add_distrib_left
tff(fact_6857_max__add__distrib__left,axiom,
    ! [X: 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,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).

% max_add_distrib_left
tff(fact_6858_max__diff__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z)) ).

% max_diff_distrib_left
tff(fact_6859_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y)),Z) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Y),Z)) ).

% max_diff_distrib_left
tff(fact_6860_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ).

% max_diff_distrib_left
tff(fact_6861_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ).

% nat_add_max_right
tff(fact_6862_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),Q3)) ).

% nat_add_max_left
tff(fact_6863_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) ).

% nat_mult_max_left
tff(fact_6864_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)) ).

% nat_mult_max_right
tff(fact_6865_nat__minus__add__max,axiom,
    ! [N: 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,N),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),M) ).

% nat_minus_add_max
tff(fact_6866_inf__sup__ord_I2_J,axiom,
    ! [X: set_nat,Y: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y)) ).

% inf_sup_ord(2)
tff(fact_6867_inf__sup__ord_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Y)) ).

% inf_sup_ord(2)
tff(fact_6868_inf__sup__ord_I2_J,axiom,
    ! [X: set_int,Y: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)),Y)) ).

% inf_sup_ord(2)
tff(fact_6869_inf__sup__ord_I2_J,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),Y)) ).

% inf_sup_ord(2)
tff(fact_6870_inf__sup__ord_I2_J,axiom,
    ! [X: nat,Y: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Y)) ).

% inf_sup_ord(2)
tff(fact_6871_inf__sup__ord_I2_J,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),Y)) ).

% inf_sup_ord(2)
tff(fact_6872_inf__sup__ord_I1_J,axiom,
    ! [X: set_nat,Y: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),X)) ).

% inf_sup_ord(1)
tff(fact_6873_inf__sup__ord_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),X)) ).

% inf_sup_ord(1)
tff(fact_6874_inf__sup__ord_I1_J,axiom,
    ! [X: set_int,Y: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)),X)) ).

% inf_sup_ord(1)
tff(fact_6875_inf__sup__ord_I1_J,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),X)) ).

% inf_sup_ord(1)
tff(fact_6876_inf__sup__ord_I1_J,axiom,
    ! [X: nat,Y: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),X)) ).

% inf_sup_ord(1)
tff(fact_6877_inf__sup__ord_I1_J,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),X)) ).

% inf_sup_ord(1)
tff(fact_6878_inf__le1,axiom,
    ! [X: set_nat,Y: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),X)) ).

% inf_le1
tff(fact_6879_inf__le1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),X)) ).

% inf_le1
tff(fact_6880_inf__le1,axiom,
    ! [X: set_int,Y: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)),X)) ).

% inf_le1
tff(fact_6881_inf__le1,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),X)) ).

% inf_le1
tff(fact_6882_inf__le1,axiom,
    ! [X: nat,Y: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),X)) ).

% inf_le1
tff(fact_6883_inf__le1,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),X)) ).

% inf_le1
tff(fact_6884_inf__le2,axiom,
    ! [X: set_nat,Y: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y)) ).

% inf_le2
tff(fact_6885_inf__le2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Y)) ).

% inf_le2
tff(fact_6886_inf__le2,axiom,
    ! [X: set_int,Y: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)),Y)) ).

% inf_le2
tff(fact_6887_inf__le2,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),Y)) ).

% inf_le2
tff(fact_6888_inf__le2,axiom,
    ! [X: nat,Y: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Y)) ).

% inf_le2
tff(fact_6889_inf__le2,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),Y)) ).

% inf_le2
tff(fact_6890_le__infE,axiom,
    ! [X: set_nat,A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3)))
     => ~ ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),A2))
         => ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),B3)) ) ) ).

% le_infE
tff(fact_6891_le__infE,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3)))
     => ~ ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),A2))
         => ~ pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),B3)) ) ) ).

% le_infE
tff(fact_6892_le__infE,axiom,
    ! [X: set_int,A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B3)))
     => ~ ( pp(aa_set_int_bool(ord_less_eq_set_int(X),A2))
         => ~ pp(aa_set_int_bool(ord_less_eq_set_int(X),B3)) ) ) ).

% le_infE
tff(fact_6893_le__infE,axiom,
    ! [X: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3)))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),A2))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),B3)) ) ) ).

% le_infE
tff(fact_6894_le__infE,axiom,
    ! [X: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3)))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),A2))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),B3)) ) ) ).

% le_infE
tff(fact_6895_le__infE,axiom,
    ! [X: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3)))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A2))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),B3)) ) ) ).

% le_infE
tff(fact_6896_le__infI,axiom,
    ! [X: set_nat,A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),B3))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3))) ) ) ).

% le_infI
tff(fact_6897_le__infI,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),A2))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),B3))
       => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3))) ) ) ).

% le_infI
tff(fact_6898_le__infI,axiom,
    ! [X: set_int,A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X),B3))
       => pp(aa_set_int_bool(ord_less_eq_set_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B3))) ) ) ).

% le_infI
tff(fact_6899_le__infI,axiom,
    ! [X: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3))) ) ) ).

% le_infI
tff(fact_6900_le__infI,axiom,
    ! [X: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3))) ) ) ).

% le_infI
tff(fact_6901_le__infI,axiom,
    ! [X: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3))) ) ) ).

% le_infI
tff(fact_6902_inf__mono,axiom,
    ! [A2: set_nat,C: set_nat,B3: set_nat,D: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),C))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),D))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C),D))) ) ) ).

% inf_mono
tff(fact_6903_inf__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),C))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(B3),D))
       => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C),D))) ) ) ).

% inf_mono
tff(fact_6904_inf__mono,axiom,
    ! [A2: set_int,C: set_int,B3: set_int,D: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),D))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B3)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,C),D))) ) ) ).

% inf_mono
tff(fact_6905_inf__mono,axiom,
    ! [A2: rat,C: rat,B3: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3)),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,C),D))) ) ) ).

% inf_mono
tff(fact_6906_inf__mono,axiom,
    ! [A2: nat,C: nat,B3: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,C),D))) ) ) ).

% inf_mono
tff(fact_6907_inf__mono,axiom,
    ! [A2: int,C: int,B3: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3)),aa_int_int(aa_int_fun_int_int(inf_inf_int,C),D))) ) ) ).

% inf_mono
tff(fact_6908_le__infI1,axiom,
    ! [A2: set_nat,X: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),X))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3)),X)) ) ).

% le_infI1
tff(fact_6909_le__infI1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),X))
     => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3)),X)) ) ).

% le_infI1
tff(fact_6910_le__infI1,axiom,
    ! [A2: set_int,X: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),X))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B3)),X)) ) ).

% le_infI1
tff(fact_6911_le__infI1,axiom,
    ! [A2: rat,X: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3)),X)) ) ).

% le_infI1
tff(fact_6912_le__infI1,axiom,
    ! [A2: nat,X: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3)),X)) ) ).

% le_infI1
tff(fact_6913_le__infI1,axiom,
    ! [A2: int,X: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3)),X)) ) ).

% le_infI1
tff(fact_6914_le__infI2,axiom,
    ! [B3: set_nat,X: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),X))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3)),X)) ) ).

% le_infI2
tff(fact_6915_le__infI2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(B3),X))
     => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3)),X)) ) ).

% le_infI2
tff(fact_6916_le__infI2,axiom,
    ! [B3: set_int,X: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),X))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B3)),X)) ) ).

% le_infI2
tff(fact_6917_le__infI2,axiom,
    ! [B3: rat,X: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3)),X)) ) ).

% le_infI2
tff(fact_6918_le__infI2,axiom,
    ! [B3: nat,X: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3)),X)) ) ).

% le_infI2
tff(fact_6919_le__infI2,axiom,
    ! [B3: int,X: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3)),X)) ) ).

% le_infI2
tff(fact_6920_inf_OorderE,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3) ) ) ).

% inf.orderE
tff(fact_6921_inf_OorderE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B3))
     => ( A2 = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3) ) ) ).

% inf.orderE
tff(fact_6922_inf_OorderE,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( A2 = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B3) ) ) ).

% inf.orderE
tff(fact_6923_inf_OorderE,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3) ) ) ).

% inf.orderE
tff(fact_6924_inf_OorderE,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3) ) ) ).

% inf.orderE
tff(fact_6925_inf_OorderE,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( A2 = aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3) ) ) ).

% inf.orderE
tff(fact_6926_inf_OorderI,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3)) ) ).

% inf.orderI
tff(fact_6927_inf_OorderI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( A2 = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3) )
     => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B3)) ) ).

% inf.orderI
tff(fact_6928_inf_OorderI,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B3) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3)) ) ).

% inf.orderI
tff(fact_6929_inf_OorderI,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3)) ) ).

% inf.orderI
tff(fact_6930_inf_OorderI,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3)) ) ).

% inf.orderI
tff(fact_6931_inf_OorderI,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ).

% inf.orderI
tff(fact_6932_inf__unique,axiom,
    ! [F: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
      ( ! [X5: set_nat,Y3: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X5),Y3)),X5))
     => ( ! [X5: set_nat,Y3: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X5),Y3)),Y3))
       => ( ! [X5: set_nat,Y3: set_nat,Z3: set_nat] :
              ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X5),Y3))
             => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X5),Z3))
               => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y3),Z3))) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_6933_inf__unique,axiom,
    ! [F: fun_se3560235952666129655at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ! [X5: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X5),Y3)),X5))
     => ( ! [X5: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X5),Y3)),Y3))
       => ( ! [X5: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X5),Y3))
             => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X5),Z3))
               => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X5),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,Y3),Z3))) ) )
         => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_6934_inf__unique,axiom,
    ! [F: fun_se6573205548028396576et_int,X: set_int,Y: set_int] :
      ( ! [X5: set_int,Y3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(F,X5),Y3)),X5))
     => ( ! [X5: set_int,Y3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(F,X5),Y3)),Y3))
       => ( ! [X5: set_int,Y3: set_int,Z3: set_int] :
              ( pp(aa_set_int_bool(ord_less_eq_set_int(X5),Y3))
             => ( pp(aa_set_int_bool(ord_less_eq_set_int(X5),Z3))
               => pp(aa_set_int_bool(ord_less_eq_set_int(X5),aa_set_int_set_int(aa_set4234024024286713313et_int(F,Y3),Z3))) ) )
         => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = aa_set_int_set_int(aa_set4234024024286713313et_int(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_6935_inf__unique,axiom,
    ! [F: fun_rat_fun_rat_rat,X: rat,Y: rat] :
      ( ! [X5: rat,Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,X5),Y3)),X5))
     => ( ! [X5: rat,Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,X5),Y3)),Y3))
       => ( ! [X5: rat,Y3: rat,Z3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Y3))
             => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Z3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(F,Y3),Z3))) ) )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_6936_inf__unique,axiom,
    ! [F: fun_nat_fun_nat_nat,X: nat,Y: nat] :
      ( ! [X5: nat,Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,X5),Y3)),X5))
     => ( ! [X5: nat,Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,X5),Y3)),Y3))
       => ( ! [X5: nat,Y3: nat,Z3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Y3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Z3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),aa_nat_nat(aa_nat_fun_nat_nat(F,Y3),Z3))) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_6937_inf__unique,axiom,
    ! [F: fun_int_fun_int_int,X: int,Y: int] :
      ( ! [X5: int,Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X5),Y3)),X5))
     => ( ! [X5: int,Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X5),Y3)),Y3))
       => ( ! [X5: int,Y3: int,Z3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),Y3))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),Z3))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),aa_int_int(aa_int_fun_int_int(F,Y3),Z3))) ) )
         => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = aa_int_int(aa_int_fun_int_int(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_6938_le__iff__inf,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_6939_le__iff__inf,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),Y))
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_6940_le__iff__inf,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_6941_le__iff__inf,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_6942_le__iff__inf,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_6943_le__iff__inf,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_6944_inf_Oabsorb1,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3) = A2 ) ) ).

% inf.absorb1
tff(fact_6945_inf_Oabsorb1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B3))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3) = A2 ) ) ).

% inf.absorb1
tff(fact_6946_inf_Oabsorb1,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B3) = A2 ) ) ).

% inf.absorb1
tff(fact_6947_inf_Oabsorb1,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3) = A2 ) ) ).

% inf.absorb1
tff(fact_6948_inf_Oabsorb1,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3) = A2 ) ) ).

% inf.absorb1
tff(fact_6949_inf_Oabsorb1,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3) = A2 ) ) ).

% inf.absorb1
tff(fact_6950_inf_Oabsorb2,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),A2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3) = B3 ) ) ).

% inf.absorb2
tff(fact_6951_inf_Oabsorb2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(B3),A2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3) = B3 ) ) ).

% inf.absorb2
tff(fact_6952_inf_Oabsorb2,axiom,
    ! [B3: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B3) = B3 ) ) ).

% inf.absorb2
tff(fact_6953_inf_Oabsorb2,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3) = B3 ) ) ).

% inf.absorb2
tff(fact_6954_inf_Oabsorb2,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3) = B3 ) ) ).

% inf.absorb2
tff(fact_6955_inf_Oabsorb2,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3) = B3 ) ) ).

% inf.absorb2
tff(fact_6956_inf__absorb1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_6957_inf__absorb1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),Y))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_6958_inf__absorb1,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_6959_inf__absorb1,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_6960_inf__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_6961_inf__absorb1,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_6962_inf__absorb2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),X))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_6963_inf__absorb2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(Y),X))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_6964_inf__absorb2,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_6965_inf__absorb2,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_6966_inf__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_6967_inf__absorb2,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_6968_inf_OboundedE,axiom,
    ! [A2: set_nat,B3: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),C)))
     => ~ ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
         => ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),C)) ) ) ).

% inf.boundedE
tff(fact_6969_inf_OboundedE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),C)))
     => ~ ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B3))
         => ~ pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),C)) ) ) ).

% inf.boundedE
tff(fact_6970_inf_OboundedE,axiom,
    ! [A2: set_int,B3: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B3),C)))
     => ~ ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
         => ~ pp(aa_set_int_bool(ord_less_eq_set_int(A2),C)) ) ) ).

% inf.boundedE
tff(fact_6971_inf_OboundedE,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B3),C)))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),C)) ) ) ).

% inf.boundedE
tff(fact_6972_inf_OboundedE,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B3),C)))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C)) ) ) ).

% inf.boundedE
tff(fact_6973_inf_OboundedE,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(inf_inf_int,B3),C)))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C)) ) ) ).

% inf.boundedE
tff(fact_6974_inf_OboundedI,axiom,
    ! [A2: set_nat,B3: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),C))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),C))) ) ) ).

% inf.boundedI
tff(fact_6975_inf_OboundedI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B3))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),C))
       => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),C))) ) ) ).

% inf.boundedI
tff(fact_6976_inf_OboundedI,axiom,
    ! [A2: set_int,B3: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B3),C))) ) ) ).

% inf.boundedI
tff(fact_6977_inf_OboundedI,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B3),C))) ) ) ).

% inf.boundedI
tff(fact_6978_inf_OboundedI,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B3),C))) ) ) ).

% inf.boundedI
tff(fact_6979_inf_OboundedI,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(inf_inf_int,B3),C))) ) ) ).

% inf.boundedI
tff(fact_6980_inf__greatest,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Z))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z))) ) ) ).

% inf_greatest
tff(fact_6981_inf__greatest,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),Y))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),Z))
       => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z))) ) ) ).

% inf_greatest
tff(fact_6982_inf__greatest,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Z))
       => pp(aa_set_int_bool(ord_less_eq_set_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Y),Z))) ) ) ).

% inf_greatest
tff(fact_6983_inf__greatest,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z))) ) ) ).

% inf_greatest
tff(fact_6984_inf__greatest,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z))) ) ) ).

% inf_greatest
tff(fact_6985_inf__greatest,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z))) ) ) ).

% inf_greatest
tff(fact_6986_inf_Oorder__iff,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
    <=> ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3) ) ) ).

% inf.order_iff
tff(fact_6987_inf_Oorder__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B3))
    <=> ( A2 = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3) ) ) ).

% inf.order_iff
tff(fact_6988_inf_Oorder__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
    <=> ( A2 = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B3) ) ) ).

% inf.order_iff
tff(fact_6989_inf_Oorder__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
    <=> ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3) ) ) ).

% inf.order_iff
tff(fact_6990_inf_Oorder__iff,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
    <=> ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3) ) ) ).

% inf.order_iff
tff(fact_6991_inf_Oorder__iff,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
    <=> ( A2 = aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3) ) ) ).

% inf.order_iff
tff(fact_6992_inf_Ocobounded1,axiom,
    ! [A2: set_nat,B3: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3)),A2)) ).

% inf.cobounded1
tff(fact_6993_inf_Ocobounded1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3)),A2)) ).

% inf.cobounded1
tff(fact_6994_inf_Ocobounded1,axiom,
    ! [A2: set_int,B3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B3)),A2)) ).

% inf.cobounded1
tff(fact_6995_inf_Ocobounded1,axiom,
    ! [A2: rat,B3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3)),A2)) ).

% inf.cobounded1
tff(fact_6996_inf_Ocobounded1,axiom,
    ! [A2: nat,B3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3)),A2)) ).

% inf.cobounded1
tff(fact_6997_inf_Ocobounded1,axiom,
    ! [A2: int,B3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3)),A2)) ).

% inf.cobounded1
tff(fact_6998_inf_Ocobounded2,axiom,
    ! [A2: set_nat,B3: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3)),B3)) ).

% inf.cobounded2
tff(fact_6999_inf_Ocobounded2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3)),B3)) ).

% inf.cobounded2
tff(fact_7000_inf_Ocobounded2,axiom,
    ! [A2: set_int,B3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B3)),B3)) ).

% inf.cobounded2
tff(fact_7001_inf_Ocobounded2,axiom,
    ! [A2: rat,B3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3)),B3)) ).

% inf.cobounded2
tff(fact_7002_inf_Ocobounded2,axiom,
    ! [A2: nat,B3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3)),B3)) ).

% inf.cobounded2
tff(fact_7003_inf_Ocobounded2,axiom,
    ! [A2: int,B3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3)),B3)) ).

% inf.cobounded2
tff(fact_7004_inf_Oabsorb__iff1,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3) = A2 ) ) ).

% inf.absorb_iff1
tff(fact_7005_inf_Oabsorb__iff1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B3))
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3) = A2 ) ) ).

% inf.absorb_iff1
tff(fact_7006_inf_Oabsorb__iff1,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B3) = A2 ) ) ).

% inf.absorb_iff1
tff(fact_7007_inf_Oabsorb__iff1,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3) = A2 ) ) ).

% inf.absorb_iff1
tff(fact_7008_inf_Oabsorb__iff1,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3) = A2 ) ) ).

% inf.absorb_iff1
tff(fact_7009_inf_Oabsorb__iff1,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3) = A2 ) ) ).

% inf.absorb_iff1
tff(fact_7010_inf_Oabsorb__iff2,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),A2))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3) = B3 ) ) ).

% inf.absorb_iff2
tff(fact_7011_inf_Oabsorb__iff2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(B3),A2))
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3) = B3 ) ) ).

% inf.absorb_iff2
tff(fact_7012_inf_Oabsorb__iff2,axiom,
    ! [B3: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B3) = B3 ) ) ).

% inf.absorb_iff2
tff(fact_7013_inf_Oabsorb__iff2,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3) = B3 ) ) ).

% inf.absorb_iff2
tff(fact_7014_inf_Oabsorb__iff2,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3) = B3 ) ) ).

% inf.absorb_iff2
tff(fact_7015_inf_Oabsorb__iff2,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3) = B3 ) ) ).

% inf.absorb_iff2
tff(fact_7016_inf_OcoboundedI1,axiom,
    ! [A2: set_nat,C: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),C))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3)),C)) ) ).

% inf.coboundedI1
tff(fact_7017_inf_OcoboundedI1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),C))
     => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3)),C)) ) ).

% inf.coboundedI1
tff(fact_7018_inf_OcoboundedI1,axiom,
    ! [A2: set_int,C: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B3)),C)) ) ).

% inf.coboundedI1
tff(fact_7019_inf_OcoboundedI1,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3)),C)) ) ).

% inf.coboundedI1
tff(fact_7020_inf_OcoboundedI1,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3)),C)) ) ).

% inf.coboundedI1
tff(fact_7021_inf_OcoboundedI1,axiom,
    ! [A2: int,C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3)),C)) ) ).

% inf.coboundedI1
tff(fact_7022_inf_OcoboundedI2,axiom,
    ! [B3: set_nat,C: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),C))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3)),C)) ) ).

% inf.coboundedI2
tff(fact_7023_inf_OcoboundedI2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(B3),C))
     => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3)),C)) ) ).

% inf.coboundedI2
tff(fact_7024_inf_OcoboundedI2,axiom,
    ! [B3: set_int,C: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),C))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B3)),C)) ) ).

% inf.coboundedI2
tff(fact_7025_inf_OcoboundedI2,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3)),C)) ) ).

% inf.coboundedI2
tff(fact_7026_inf_OcoboundedI2,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3)),C)) ) ).

% inf.coboundedI2
tff(fact_7027_inf_OcoboundedI2,axiom,
    ! [B3: int,C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3)),C)) ) ).

% inf.coboundedI2
tff(fact_7028_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat,X: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y))) ).

% inf_sup_ord(4)
tff(fact_7029_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(Y,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_7030_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(Y,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_7031_inf__sup__ord_I4_J,axiom,
    ! [Y: set_int,X: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(Y),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y))) ).

% inf_sup_ord(4)
tff(fact_7032_inf__sup__ord_I4_J,axiom,
    ! [Y: rat,X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y))) ).

% inf_sup_ord(4)
tff(fact_7033_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y))) ).

% inf_sup_ord(4)
tff(fact_7034_inf__sup__ord_I4_J,axiom,
    ! [Y: int,X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y))) ).

% inf_sup_ord(4)
tff(fact_7035_inf__sup__ord_I3_J,axiom,
    ! [X: set_nat,Y: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y))) ).

% inf_sup_ord(3)
tff(fact_7036_inf__sup__ord_I3_J,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(X,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_7037_inf__sup__ord_I3_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_7038_inf__sup__ord_I3_J,axiom,
    ! [X: set_int,Y: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y))) ).

% inf_sup_ord(3)
tff(fact_7039_inf__sup__ord_I3_J,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y))) ).

% inf_sup_ord(3)
tff(fact_7040_inf__sup__ord_I3_J,axiom,
    ! [X: nat,Y: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y))) ).

% inf_sup_ord(3)
tff(fact_7041_inf__sup__ord_I3_J,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y))) ).

% inf_sup_ord(3)
tff(fact_7042_le__supE,axiom,
    ! [A2: set_nat,B3: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3)),X))
     => ~ ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),X))
         => ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),X)) ) ) ).

% le_supE
tff(fact_7043_le__supE,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3),X)
     => ~ ( ord_le3000389064537975527at_nat(A2,X)
         => ~ ord_le3000389064537975527at_nat(B3,X) ) ) ).

% le_supE
tff(fact_7044_le__supE,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3),X)
     => ~ ( ord_le1268244103169919719at_nat(A2,X)
         => ~ ord_le1268244103169919719at_nat(B3,X) ) ) ).

% le_supE
tff(fact_7045_le__supE,axiom,
    ! [A2: set_int,B3: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3)),X))
     => ~ ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),X))
         => ~ pp(aa_set_int_bool(ord_less_eq_set_int(B3),X)) ) ) ).

% le_supE
tff(fact_7046_le__supE,axiom,
    ! [A2: rat,B3: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3)),X))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),X))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),X)) ) ) ).

% le_supE
tff(fact_7047_le__supE,axiom,
    ! [A2: nat,B3: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3)),X))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),X))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),X)) ) ) ).

% le_supE
tff(fact_7048_le__supE,axiom,
    ! [A2: int,B3: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3)),X))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),X))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),X)) ) ) ).

% le_supE
tff(fact_7049_le__supI,axiom,
    ! [A2: set_nat,X: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),X))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),X))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3)),X)) ) ) ).

% le_supI
tff(fact_7050_le__supI,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A2,X)
     => ( ord_le3000389064537975527at_nat(B3,X)
       => ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3),X) ) ) ).

% le_supI
tff(fact_7051_le__supI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A2,X)
     => ( ord_le1268244103169919719at_nat(B3,X)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3),X) ) ) ).

% le_supI
tff(fact_7052_le__supI,axiom,
    ! [A2: set_int,X: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),X))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),X))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3)),X)) ) ) ).

% le_supI
tff(fact_7053_le__supI,axiom,
    ! [A2: rat,X: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3)),X)) ) ) ).

% le_supI
tff(fact_7054_le__supI,axiom,
    ! [A2: nat,X: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),X))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3)),X)) ) ) ).

% le_supI
tff(fact_7055_le__supI,axiom,
    ! [A2: int,X: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),X))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3)),X)) ) ) ).

% le_supI
tff(fact_7056_sup__ge1,axiom,
    ! [X: set_nat,Y: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y))) ).

% sup_ge1
tff(fact_7057_sup__ge1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(X,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)) ).

% sup_ge1
tff(fact_7058_sup__ge1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% sup_ge1
tff(fact_7059_sup__ge1,axiom,
    ! [X: set_int,Y: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y))) ).

% sup_ge1
tff(fact_7060_sup__ge1,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y))) ).

% sup_ge1
tff(fact_7061_sup__ge1,axiom,
    ! [X: nat,Y: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y))) ).

% sup_ge1
tff(fact_7062_sup__ge1,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y))) ).

% sup_ge1
tff(fact_7063_sup__ge2,axiom,
    ! [Y: set_nat,X: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y))) ).

% sup_ge2
tff(fact_7064_sup__ge2,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(Y,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)) ).

% sup_ge2
tff(fact_7065_sup__ge2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(Y,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% sup_ge2
tff(fact_7066_sup__ge2,axiom,
    ! [Y: set_int,X: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(Y),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y))) ).

% sup_ge2
tff(fact_7067_sup__ge2,axiom,
    ! [Y: rat,X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y))) ).

% sup_ge2
tff(fact_7068_sup__ge2,axiom,
    ! [Y: nat,X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y))) ).

% sup_ge2
tff(fact_7069_sup__ge2,axiom,
    ! [Y: int,X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y))) ).

% sup_ge2
tff(fact_7070_le__supI1,axiom,
    ! [X: set_nat,A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3))) ) ).

% le_supI1
tff(fact_7071_le__supI1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(X,A2)
     => ord_le3000389064537975527at_nat(X,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3)) ) ).

% le_supI1
tff(fact_7072_le__supI1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(X,A2)
     => ord_le1268244103169919719at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3)) ) ).

% le_supI1
tff(fact_7073_le__supI1,axiom,
    ! [X: set_int,A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3))) ) ).

% le_supI1
tff(fact_7074_le__supI1,axiom,
    ! [X: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3))) ) ).

% le_supI1
tff(fact_7075_le__supI1,axiom,
    ! [X: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3))) ) ).

% le_supI1
tff(fact_7076_le__supI1,axiom,
    ! [X: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3))) ) ).

% le_supI1
tff(fact_7077_le__supI2,axiom,
    ! [X: set_nat,B3: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),B3))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3))) ) ).

% le_supI2
tff(fact_7078_le__supI2,axiom,
    ! [X: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(X,B3)
     => ord_le3000389064537975527at_nat(X,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3)) ) ).

% le_supI2
tff(fact_7079_le__supI2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(X,B3)
     => ord_le1268244103169919719at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3)) ) ).

% le_supI2
tff(fact_7080_le__supI2,axiom,
    ! [X: set_int,B3: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),B3))
     => pp(aa_set_int_bool(ord_less_eq_set_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3))) ) ).

% le_supI2
tff(fact_7081_le__supI2,axiom,
    ! [X: rat,B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3))) ) ).

% le_supI2
tff(fact_7082_le__supI2,axiom,
    ! [X: nat,B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),B3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3))) ) ).

% le_supI2
tff(fact_7083_le__supI2,axiom,
    ! [X: int,B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3))) ) ).

% le_supI2
tff(fact_7084_sup_Omono,axiom,
    ! [C: set_nat,A2: set_nat,D: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,D),B3))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),D)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3))) ) ) ).

% sup.mono
tff(fact_7085_sup_Omono,axiom,
    ! [C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,D: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(C,A2)
     => ( ord_le3000389064537975527at_nat(D,B3)
       => ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,C),D),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3)) ) ) ).

% sup.mono
tff(fact_7086_sup_Omono,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(C,A2)
     => ( ord_le1268244103169919719at_nat(D,B3)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C),D),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3)) ) ) ).

% sup.mono
tff(fact_7087_sup_Omono,axiom,
    ! [C: set_int,A2: set_int,D: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(C),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(D),B3))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,C),D)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3))) ) ) ).

% sup.mono
tff(fact_7088_sup_Omono,axiom,
    ! [C: rat,A2: rat,D: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D),B3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,C),D)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3))) ) ) ).

% sup.mono
tff(fact_7089_sup_Omono,axiom,
    ! [C: nat,A2: nat,D: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3))) ) ) ).

% sup.mono
tff(fact_7090_sup_Omono,axiom,
    ! [C: int,A2: int,D: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),B3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,C),D)),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3))) ) ) ).

% sup.mono
tff(fact_7091_sup__mono,axiom,
    ! [A2: set_nat,C: set_nat,B3: set_nat,D: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),C))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),D))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),D))) ) ) ).

% sup_mono
tff(fact_7092_sup__mono,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,D: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A2,C)
     => ( ord_le3000389064537975527at_nat(B3,D)
       => ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,C),D)) ) ) ).

% sup_mono
tff(fact_7093_sup__mono,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A2,C)
     => ( ord_le1268244103169919719at_nat(B3,D)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C),D)) ) ) ).

% sup_mono
tff(fact_7094_sup__mono,axiom,
    ! [A2: set_int,C: set_int,B3: set_int,D: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),D))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,C),D))) ) ) ).

% sup_mono
tff(fact_7095_sup__mono,axiom,
    ! [A2: rat,C: rat,B3: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,C),D))) ) ) ).

% sup_mono
tff(fact_7096_sup__mono,axiom,
    ! [A2: nat,C: nat,B3: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C),D))) ) ) ).

% sup_mono
tff(fact_7097_sup__mono,axiom,
    ! [A2: int,C: int,B3: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3)),aa_int_int(aa_int_fun_int_int(sup_sup_int,C),D))) ) ) ).

% sup_mono
tff(fact_7098_sup__least,axiom,
    ! [Y: set_nat,X: set_nat,Z: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),X))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Z),X))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),X)) ) ) ).

% sup_least
tff(fact_7099_sup__least,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(Y,X)
     => ( ord_le3000389064537975527at_nat(Z,X)
       => ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),Z),X) ) ) ).

% sup_least
tff(fact_7100_sup__least,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(Y,X)
     => ( ord_le1268244103169919719at_nat(Z,X)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z),X) ) ) ).

% sup_least
tff(fact_7101_sup__least,axiom,
    ! [Y: set_int,X: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Z),X))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Y),Z)),X)) ) ) ).

% sup_least
tff(fact_7102_sup__least,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Y),Z)),X)) ) ) ).

% sup_least
tff(fact_7103_sup__least,axiom,
    ! [Y: nat,X: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)),X)) ) ) ).

% sup_least
tff(fact_7104_sup__least,axiom,
    ! [Y: int,X: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)),X)) ) ) ).

% sup_least
tff(fact_7105_le__iff__sup,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_7106_le__iff__sup,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(X,Y)
    <=> ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_7107_le__iff__sup,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(X,Y)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_7108_le__iff__sup,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_7109_le__iff__sup,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_7110_le__iff__sup,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_7111_le__iff__sup,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_7112_sup_OorderE,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),A2))
     => ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) ) ) ).

% sup.orderE
tff(fact_7113_sup_OorderE,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(B3,A2)
     => ( A2 = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3) ) ) ).

% sup.orderE
tff(fact_7114_sup_OorderE,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B3,A2)
     => ( A2 = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3) ) ) ).

% sup.orderE
tff(fact_7115_sup_OorderE,axiom,
    ! [B3: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
     => ( A2 = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3) ) ) ).

% sup.orderE
tff(fact_7116_sup_OorderE,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
     => ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3) ) ) ).

% sup.orderE
tff(fact_7117_sup_OorderE,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
     => ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3) ) ) ).

% sup.orderE
tff(fact_7118_sup_OorderE,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
     => ( A2 = aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3) ) ) ).

% sup.orderE
tff(fact_7119_sup_OorderI,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),A2)) ) ).

% sup.orderI
tff(fact_7120_sup_OorderI,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( A2 = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3) )
     => ord_le3000389064537975527at_nat(B3,A2) ) ).

% sup.orderI
tff(fact_7121_sup_OorderI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( A2 = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3) )
     => ord_le1268244103169919719at_nat(B3,A2) ) ).

% sup.orderI
tff(fact_7122_sup_OorderI,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2)) ) ).

% sup.orderI
tff(fact_7123_sup_OorderI,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2)) ) ).

% sup.orderI
tff(fact_7124_sup_OorderI,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2)) ) ).

% sup.orderI
tff(fact_7125_sup_OorderI,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2)) ) ).

% sup.orderI
tff(fact_7126_sup__unique,axiom,
    ! [F: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
      ( ! [X5: set_nat,Y3: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X5),Y3)))
     => ( ! [X5: set_nat,Y3: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X5),Y3)))
       => ( ! [X5: set_nat,Y3: set_nat,Z3: set_nat] :
              ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y3),X5))
             => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Z3),X5))
               => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y3),Z3)),X5)) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_7127_sup__unique,axiom,
    ! [F: fun_se8464659365716347869at_nat,X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ! [X5: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(X5,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(F,X5),Y3))
     => ( ! [X5: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(Y3,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(F,X5),Y3))
       => ( ! [X5: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z3: set_Pr8693737435421807431at_nat] :
              ( ord_le3000389064537975527at_nat(Y3,X5)
             => ( ord_le3000389064537975527at_nat(Z3,X5)
               => ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(F,Y3),Z3),X5) ) )
         => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_7128_sup__unique,axiom,
    ! [F: fun_se177437144654672861at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ! [X5: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(X5,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,X5),Y3))
     => ( ! [X5: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(Y3,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,X5),Y3))
       => ( ! [X5: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
              ( ord_le1268244103169919719at_nat(Y3,X5)
             => ( ord_le1268244103169919719at_nat(Z3,X5)
               => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,Y3),Z3),X5) ) )
         => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_7129_sup__unique,axiom,
    ! [F: fun_se6573205548028396576et_int,X: set_int,Y: set_int] :
      ( ! [X5: set_int,Y3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(X5),aa_set_int_set_int(aa_set4234024024286713313et_int(F,X5),Y3)))
     => ( ! [X5: set_int,Y3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(Y3),aa_set_int_set_int(aa_set4234024024286713313et_int(F,X5),Y3)))
       => ( ! [X5: set_int,Y3: set_int,Z3: set_int] :
              ( pp(aa_set_int_bool(ord_less_eq_set_int(Y3),X5))
             => ( pp(aa_set_int_bool(ord_less_eq_set_int(Z3),X5))
               => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(F,Y3),Z3)),X5)) ) )
         => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = aa_set_int_set_int(aa_set4234024024286713313et_int(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_7130_sup__unique,axiom,
    ! [F: fun_rat_fun_rat_rat,X: rat,Y: rat] :
      ( ! [X5: rat,Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(F,X5),Y3)))
     => ( ! [X5: rat,Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),aa_rat_rat(aa_rat_fun_rat_rat(F,X5),Y3)))
       => ( ! [X5: rat,Y3: rat,Z3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X5))
             => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z3),X5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,Y3),Z3)),X5)) ) )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_7131_sup__unique,axiom,
    ! [F: fun_nat_fun_nat_nat,X: nat,Y: nat] :
      ( ! [X5: nat,Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),aa_nat_nat(aa_nat_fun_nat_nat(F,X5),Y3)))
     => ( ! [X5: nat,Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(F,X5),Y3)))
       => ( ! [X5: nat,Y3: nat,Z3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X5))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z3),X5))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,Y3),Z3)),X5)) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_7132_sup__unique,axiom,
    ! [F: fun_int_fun_int_int,X: int,Y: int] :
      ( ! [X5: int,Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),aa_int_int(aa_int_fun_int_int(F,X5),Y3)))
     => ( ! [X5: int,Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),aa_int_int(aa_int_fun_int_int(F,X5),Y3)))
       => ( ! [X5: int,Y3: int,Z3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X5))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z3),X5))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,Y3),Z3)),X5)) ) )
         => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = aa_int_int(aa_int_fun_int_int(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_7133_sup_Oabsorb1,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),A2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) = A2 ) ) ).

% sup.absorb1
tff(fact_7134_sup_Oabsorb1,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(B3,A2)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3) = A2 ) ) ).

% sup.absorb1
tff(fact_7135_sup_Oabsorb1,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B3,A2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3) = A2 ) ) ).

% sup.absorb1
tff(fact_7136_sup_Oabsorb1,axiom,
    ! [B3: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3) = A2 ) ) ).

% sup.absorb1
tff(fact_7137_sup_Oabsorb1,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3) = A2 ) ) ).

% sup.absorb1
tff(fact_7138_sup_Oabsorb1,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3) = A2 ) ) ).

% sup.absorb1
tff(fact_7139_sup_Oabsorb1,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3) = A2 ) ) ).

% sup.absorb1
tff(fact_7140_sup_Oabsorb2,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) = B3 ) ) ).

% sup.absorb2
tff(fact_7141_sup_Oabsorb2,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A2,B3)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3) = B3 ) ) ).

% sup.absorb2
tff(fact_7142_sup_Oabsorb2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A2,B3)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3) = B3 ) ) ).

% sup.absorb2
tff(fact_7143_sup_Oabsorb2,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3) = B3 ) ) ).

% sup.absorb2
tff(fact_7144_sup_Oabsorb2,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3) = B3 ) ) ).

% sup.absorb2
tff(fact_7145_sup_Oabsorb2,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3) = B3 ) ) ).

% sup.absorb2
tff(fact_7146_sup_Oabsorb2,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3) = B3 ) ) ).

% sup.absorb2
tff(fact_7147_sup__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),X))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_7148_sup__absorb1,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(Y,X)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_7149_sup__absorb1,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(Y,X)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_7150_sup__absorb1,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_7151_sup__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_7152_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_7153_sup__absorb1,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_7154_sup__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_7155_sup__absorb2,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(X,Y)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_7156_sup__absorb2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(X,Y)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_7157_sup__absorb2,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_7158_sup__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_7159_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_7160_sup__absorb2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_7161_sup_OboundedE,axiom,
    ! [B3: set_nat,C: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),C)),A2))
     => ~ ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),A2))
         => ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),A2)) ) ) ).

% sup.boundedE
tff(fact_7162_sup_OboundedE,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B3),C),A2)
     => ~ ( ord_le3000389064537975527at_nat(B3,A2)
         => ~ ord_le3000389064537975527at_nat(C,A2) ) ) ).

% sup.boundedE
tff(fact_7163_sup_OboundedE,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B3),C),A2)
     => ~ ( ord_le1268244103169919719at_nat(B3,A2)
         => ~ ord_le1268244103169919719at_nat(C,A2) ) ) ).

% sup.boundedE
tff(fact_7164_sup_OboundedE,axiom,
    ! [B3: set_int,C: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B3),C)),A2))
     => ~ ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
         => ~ pp(aa_set_int_bool(ord_less_eq_set_int(C),A2)) ) ) ).

% sup.boundedE
tff(fact_7165_sup_OboundedE,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B3),C)),A2))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A2)) ) ) ).

% sup.boundedE
tff(fact_7166_sup_OboundedE,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B3),C)),A2))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2)) ) ) ).

% sup.boundedE
tff(fact_7167_sup_OboundedE,axiom,
    ! [B3: int,C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B3),C)),A2))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2)) ) ) ).

% sup.boundedE
tff(fact_7168_sup_OboundedI,axiom,
    ! [B3: set_nat,A2: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),A2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),C)),A2)) ) ) ).

% sup.boundedI
tff(fact_7169_sup_OboundedI,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(B3,A2)
     => ( ord_le3000389064537975527at_nat(C,A2)
       => ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B3),C),A2) ) ) ).

% sup.boundedI
tff(fact_7170_sup_OboundedI,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B3,A2)
     => ( ord_le1268244103169919719at_nat(C,A2)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B3),C),A2) ) ) ).

% sup.boundedI
tff(fact_7171_sup_OboundedI,axiom,
    ! [B3: set_int,A2: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(C),A2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B3),C)),A2)) ) ) ).

% sup.boundedI
tff(fact_7172_sup_OboundedI,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B3),C)),A2)) ) ) ).

% sup.boundedI
tff(fact_7173_sup_OboundedI,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B3),C)),A2)) ) ) ).

% sup.boundedI
tff(fact_7174_sup_OboundedI,axiom,
    ! [B3: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B3),C)),A2)) ) ) ).

% sup.boundedI
tff(fact_7175_sup_Oorder__iff,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),A2))
    <=> ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) ) ) ).

% sup.order_iff
tff(fact_7176_sup_Oorder__iff,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(B3,A2)
    <=> ( A2 = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3) ) ) ).

% sup.order_iff
tff(fact_7177_sup_Oorder__iff,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B3,A2)
    <=> ( A2 = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3) ) ) ).

% sup.order_iff
tff(fact_7178_sup_Oorder__iff,axiom,
    ! [B3: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
    <=> ( A2 = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3) ) ) ).

% sup.order_iff
tff(fact_7179_sup_Oorder__iff,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
    <=> ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3) ) ) ).

% sup.order_iff
tff(fact_7180_sup_Oorder__iff,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
    <=> ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3) ) ) ).

% sup.order_iff
tff(fact_7181_sup_Oorder__iff,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
    <=> ( A2 = aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3) ) ) ).

% sup.order_iff
tff(fact_7182_sup_Ocobounded1,axiom,
    ! [A2: set_nat,B3: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3))) ).

% sup.cobounded1
tff(fact_7183_sup_Ocobounded1,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(A2,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3)) ).

% sup.cobounded1
tff(fact_7184_sup_Ocobounded1,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(A2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3)) ).

% sup.cobounded1
tff(fact_7185_sup_Ocobounded1,axiom,
    ! [A2: set_int,B3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3))) ).

% sup.cobounded1
tff(fact_7186_sup_Ocobounded1,axiom,
    ! [A2: rat,B3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3))) ).

% sup.cobounded1
tff(fact_7187_sup_Ocobounded1,axiom,
    ! [A2: nat,B3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3))) ).

% sup.cobounded1
tff(fact_7188_sup_Ocobounded1,axiom,
    ! [A2: int,B3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3))) ).

% sup.cobounded1
tff(fact_7189_sup_Ocobounded2,axiom,
    ! [B3: set_nat,A2: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3))) ).

% sup.cobounded2
tff(fact_7190_sup_Ocobounded2,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(B3,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3)) ).

% sup.cobounded2
tff(fact_7191_sup_Ocobounded2,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(B3,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3)) ).

% sup.cobounded2
tff(fact_7192_sup_Ocobounded2,axiom,
    ! [B3: set_int,A2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(B3),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3))) ).

% sup.cobounded2
tff(fact_7193_sup_Ocobounded2,axiom,
    ! [B3: rat,A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3))) ).

% sup.cobounded2
tff(fact_7194_sup_Ocobounded2,axiom,
    ! [B3: nat,A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3))) ).

% sup.cobounded2
tff(fact_7195_sup_Ocobounded2,axiom,
    ! [B3: int,A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3))) ).

% sup.cobounded2
tff(fact_7196_sup_Oabsorb__iff1,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),A2))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) = A2 ) ) ).

% sup.absorb_iff1
tff(fact_7197_sup_Oabsorb__iff1,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(B3,A2)
    <=> ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3) = A2 ) ) ).

% sup.absorb_iff1
tff(fact_7198_sup_Oabsorb__iff1,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B3,A2)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3) = A2 ) ) ).

% sup.absorb_iff1
tff(fact_7199_sup_Oabsorb__iff1,axiom,
    ! [B3: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3) = A2 ) ) ).

% sup.absorb_iff1
tff(fact_7200_sup_Oabsorb__iff1,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A2))
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3) = A2 ) ) ).

% sup.absorb_iff1
tff(fact_7201_sup_Oabsorb__iff1,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A2))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3) = A2 ) ) ).

% sup.absorb_iff1
tff(fact_7202_sup_Oabsorb__iff1,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2))
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3) = A2 ) ) ).

% sup.absorb_iff1
tff(fact_7203_sup_Oabsorb__iff2,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) = B3 ) ) ).

% sup.absorb_iff2
tff(fact_7204_sup_Oabsorb__iff2,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(A2,B3)
    <=> ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3) = B3 ) ) ).

% sup.absorb_iff2
tff(fact_7205_sup_Oabsorb__iff2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A2,B3)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3) = B3 ) ) ).

% sup.absorb_iff2
tff(fact_7206_sup_Oabsorb__iff2,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3) = B3 ) ) ).

% sup.absorb_iff2
tff(fact_7207_sup_Oabsorb__iff2,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B3))
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3) = B3 ) ) ).

% sup.absorb_iff2
tff(fact_7208_sup_Oabsorb__iff2,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3) = B3 ) ) ).

% sup.absorb_iff2
tff(fact_7209_sup_Oabsorb__iff2,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3) = B3 ) ) ).

% sup.absorb_iff2
tff(fact_7210_sup_OcoboundedI1,axiom,
    ! [C: set_nat,A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3))) ) ).

% sup.coboundedI1
tff(fact_7211_sup_OcoboundedI1,axiom,
    ! [C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(C,A2)
     => ord_le3000389064537975527at_nat(C,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3)) ) ).

% sup.coboundedI1
tff(fact_7212_sup_OcoboundedI1,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(C,A2)
     => ord_le1268244103169919719at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3)) ) ).

% sup.coboundedI1
tff(fact_7213_sup_OcoboundedI1,axiom,
    ! [C: set_int,A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(C),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3))) ) ).

% sup.coboundedI1
tff(fact_7214_sup_OcoboundedI1,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3))) ) ).

% sup.coboundedI1
tff(fact_7215_sup_OcoboundedI1,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3))) ) ).

% sup.coboundedI1
tff(fact_7216_sup_OcoboundedI1,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3))) ) ).

% sup.coboundedI1
tff(fact_7217_sup_OcoboundedI2,axiom,
    ! [C: set_nat,B3: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),B3))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3))) ) ).

% sup.coboundedI2
tff(fact_7218_sup_OcoboundedI2,axiom,
    ! [C: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ord_le3000389064537975527at_nat(C,B3)
     => ord_le3000389064537975527at_nat(C,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3)) ) ).

% sup.coboundedI2
tff(fact_7219_sup_OcoboundedI2,axiom,
    ! [C: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(C,B3)
     => ord_le1268244103169919719at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3)) ) ).

% sup.coboundedI2
tff(fact_7220_sup_OcoboundedI2,axiom,
    ! [C: set_int,B3: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(C),B3))
     => pp(aa_set_int_bool(ord_less_eq_set_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3))) ) ).

% sup.coboundedI2
tff(fact_7221_sup_OcoboundedI2,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3))) ) ).

% sup.coboundedI2
tff(fact_7222_sup_OcoboundedI2,axiom,
    ! [C: nat,B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3))) ) ).

% sup.coboundedI2
tff(fact_7223_sup_OcoboundedI2,axiom,
    ! [C: int,B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3))) ) ).

% sup.coboundedI2
tff(fact_7224_less__infI1,axiom,
    ! [A2: set_nat,X: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),X))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3)),X)) ) ).

% less_infI1
tff(fact_7225_less__infI1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),X))
     => pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3)),X)) ) ).

% less_infI1
tff(fact_7226_less__infI1,axiom,
    ! [A2: real,X: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A2),B3)),X)) ) ).

% less_infI1
tff(fact_7227_less__infI1,axiom,
    ! [A2: rat,X: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3)),X)) ) ).

% less_infI1
tff(fact_7228_less__infI1,axiom,
    ! [A2: nat,X: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3)),X)) ) ).

% less_infI1
tff(fact_7229_less__infI1,axiom,
    ! [A2: int,X: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3)),X)) ) ).

% less_infI1
tff(fact_7230_less__infI2,axiom,
    ! [B3: set_nat,X: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B3),X))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3)),X)) ) ).

% less_infI2
tff(fact_7231_less__infI2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(B3),X))
     => pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3)),X)) ) ).

% less_infI2
tff(fact_7232_less__infI2,axiom,
    ! [B3: real,X: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A2),B3)),X)) ) ).

% less_infI2
tff(fact_7233_less__infI2,axiom,
    ! [B3: rat,X: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3)),X)) ) ).

% less_infI2
tff(fact_7234_less__infI2,axiom,
    ! [B3: nat,X: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3)),X)) ) ).

% less_infI2
tff(fact_7235_less__infI2,axiom,
    ! [B3: int,X: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3)),X)) ) ).

% less_infI2
tff(fact_7236_inf_Oabsorb3,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3) = A2 ) ) ).

% inf.absorb3
tff(fact_7237_inf_Oabsorb3,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),B3))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3) = A2 ) ) ).

% inf.absorb3
tff(fact_7238_inf_Oabsorb3,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A2),B3) = A2 ) ) ).

% inf.absorb3
tff(fact_7239_inf_Oabsorb3,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3) = A2 ) ) ).

% inf.absorb3
tff(fact_7240_inf_Oabsorb3,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3) = A2 ) ) ).

% inf.absorb3
tff(fact_7241_inf_Oabsorb3,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3) = A2 ) ) ).

% inf.absorb3
tff(fact_7242_inf_Oabsorb4,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B3),A2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3) = B3 ) ) ).

% inf.absorb4
tff(fact_7243_inf_Oabsorb4,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(B3),A2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3) = B3 ) ) ).

% inf.absorb4
tff(fact_7244_inf_Oabsorb4,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A2),B3) = B3 ) ) ).

% inf.absorb4
tff(fact_7245_inf_Oabsorb4,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3) = B3 ) ) ).

% inf.absorb4
tff(fact_7246_inf_Oabsorb4,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3) = B3 ) ) ).

% inf.absorb4
tff(fact_7247_inf_Oabsorb4,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3) = B3 ) ) ).

% inf.absorb4
tff(fact_7248_inf_Ostrict__boundedE,axiom,
    ! [A2: set_nat,B3: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B3),C)))
     => ~ ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B3))
         => ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),C)) ) ) ).

% inf.strict_boundedE
tff(fact_7249_inf_Ostrict__boundedE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B3),C)))
     => ~ ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),B3))
         => ~ pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),C)) ) ) ).

% inf.strict_boundedE
tff(fact_7250_inf_Ostrict__boundedE,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,B3),C)))
     => ~ ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
         => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C)) ) ) ).

% inf.strict_boundedE
tff(fact_7251_inf_Ostrict__boundedE,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B3),C)))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),C)) ) ) ).

% inf.strict_boundedE
tff(fact_7252_inf_Ostrict__boundedE,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B3),C)))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C)) ) ) ).

% inf.strict_boundedE
tff(fact_7253_inf_Ostrict__boundedE,axiom,
    ! [A2: int,B3: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(inf_inf_int,B3),C)))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C)) ) ) ).

% inf.strict_boundedE
tff(fact_7254_inf_Ostrict__order__iff,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B3))
    <=> ( ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% inf.strict_order_iff
tff(fact_7255_inf_Ostrict__order__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),B3))
    <=> ( ( A2 = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% inf.strict_order_iff
tff(fact_7256_inf_Ostrict__order__iff,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
    <=> ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% inf.strict_order_iff
tff(fact_7257_inf_Ostrict__order__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
    <=> ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% inf.strict_order_iff
tff(fact_7258_inf_Ostrict__order__iff,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
    <=> ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% inf.strict_order_iff
tff(fact_7259_inf_Ostrict__order__iff,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
    <=> ( ( A2 = aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% inf.strict_order_iff
tff(fact_7260_inf_Ostrict__coboundedI1,axiom,
    ! [A2: set_nat,C: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),C))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3)),C)) ) ).

% inf.strict_coboundedI1
tff(fact_7261_inf_Ostrict__coboundedI1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),C))
     => pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3)),C)) ) ).

% inf.strict_coboundedI1
tff(fact_7262_inf_Ostrict__coboundedI1,axiom,
    ! [A2: real,C: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A2),B3)),C)) ) ).

% inf.strict_coboundedI1
tff(fact_7263_inf_Ostrict__coboundedI1,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3)),C)) ) ).

% inf.strict_coboundedI1
tff(fact_7264_inf_Ostrict__coboundedI1,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3)),C)) ) ).

% inf.strict_coboundedI1
tff(fact_7265_inf_Ostrict__coboundedI1,axiom,
    ! [A2: int,C: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3)),C)) ) ).

% inf.strict_coboundedI1
tff(fact_7266_inf_Ostrict__coboundedI2,axiom,
    ! [B3: set_nat,C: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B3),C))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B3)),C)) ) ).

% inf.strict_coboundedI2
tff(fact_7267_inf_Ostrict__coboundedI2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(B3),C))
     => pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B3)),C)) ) ).

% inf.strict_coboundedI2
tff(fact_7268_inf_Ostrict__coboundedI2,axiom,
    ! [B3: real,C: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A2),B3)),C)) ) ).

% inf.strict_coboundedI2
tff(fact_7269_inf_Ostrict__coboundedI2,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A2),B3)),C)) ) ).

% inf.strict_coboundedI2
tff(fact_7270_inf_Ostrict__coboundedI2,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B3)),C)) ) ).

% inf.strict_coboundedI2
tff(fact_7271_inf_Ostrict__coboundedI2,axiom,
    ! [B3: int,C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B3)),C)) ) ).

% inf.strict_coboundedI2
tff(fact_7272_less__supI1,axiom,
    ! [X: set_nat,A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3))) ) ).

% less_supI1
tff(fact_7273_less__supI1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ord_le6428140832669894131at_nat(X,A2)
     => ord_le6428140832669894131at_nat(X,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3)) ) ).

% less_supI1
tff(fact_7274_less__supI1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(X,A2)
     => ord_le2604355607129572851at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3)) ) ).

% less_supI1
tff(fact_7275_less__supI1,axiom,
    ! [X: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A2),B3))) ) ).

% less_supI1
tff(fact_7276_less__supI1,axiom,
    ! [X: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3))) ) ).

% less_supI1
tff(fact_7277_less__supI1,axiom,
    ! [X: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3))) ) ).

% less_supI1
tff(fact_7278_less__supI1,axiom,
    ! [X: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3))) ) ).

% less_supI1
tff(fact_7279_less__supI2,axiom,
    ! [X: set_nat,B3: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X),B3))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3))) ) ).

% less_supI2
tff(fact_7280_less__supI2,axiom,
    ! [X: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ord_le6428140832669894131at_nat(X,B3)
     => ord_le6428140832669894131at_nat(X,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3)) ) ).

% less_supI2
tff(fact_7281_less__supI2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(X,B3)
     => ord_le2604355607129572851at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3)) ) ).

% less_supI2
tff(fact_7282_less__supI2,axiom,
    ! [X: real,B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A2),B3))) ) ).

% less_supI2
tff(fact_7283_less__supI2,axiom,
    ! [X: rat,B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3))) ) ).

% less_supI2
tff(fact_7284_less__supI2,axiom,
    ! [X: nat,B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),B3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3))) ) ).

% less_supI2
tff(fact_7285_less__supI2,axiom,
    ! [X: int,B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3))) ) ).

% less_supI2
tff(fact_7286_sup_Oabsorb3,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B3),A2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) = A2 ) ) ).

% sup.absorb3
tff(fact_7287_sup_Oabsorb3,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ord_le6428140832669894131at_nat(B3,A2)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3) = A2 ) ) ).

% sup.absorb3
tff(fact_7288_sup_Oabsorb3,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(B3,A2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3) = A2 ) ) ).

% sup.absorb3
tff(fact_7289_sup_Oabsorb3,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A2),B3) = A2 ) ) ).

% sup.absorb3
tff(fact_7290_sup_Oabsorb3,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3) = A2 ) ) ).

% sup.absorb3
tff(fact_7291_sup_Oabsorb3,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3) = A2 ) ) ).

% sup.absorb3
tff(fact_7292_sup_Oabsorb3,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3) = A2 ) ) ).

% sup.absorb3
tff(fact_7293_sup_Oabsorb4,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) = B3 ) ) ).

% sup.absorb4
tff(fact_7294_sup_Oabsorb4,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ord_le6428140832669894131at_nat(A2,B3)
     => ( aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3) = B3 ) ) ).

% sup.absorb4
tff(fact_7295_sup_Oabsorb4,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(A2,B3)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3) = B3 ) ) ).

% sup.absorb4
tff(fact_7296_sup_Oabsorb4,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A2),B3) = B3 ) ) ).

% sup.absorb4
tff(fact_7297_sup_Oabsorb4,axiom,
    ! [A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B3))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3) = B3 ) ) ).

% sup.absorb4
tff(fact_7298_sup_Oabsorb4,axiom,
    ! [A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3) = B3 ) ) ).

% sup.absorb4
tff(fact_7299_sup_Oabsorb4,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3) = B3 ) ) ).

% sup.absorb4
tff(fact_7300_sup_Ostrict__boundedE,axiom,
    ! [B3: set_nat,C: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),C)),A2))
     => ~ ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B3),A2))
         => ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,C),A2)) ) ) ).

% sup.strict_boundedE
tff(fact_7301_sup_Ostrict__boundedE,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ord_le6428140832669894131at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,B3),C),A2)
     => ~ ( ord_le6428140832669894131at_nat(B3,A2)
         => ~ ord_le6428140832669894131at_nat(C,A2) ) ) ).

% sup.strict_boundedE
tff(fact_7302_sup_Ostrict__boundedE,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B3),C),A2)
     => ~ ( ord_le2604355607129572851at_nat(B3,A2)
         => ~ ord_le2604355607129572851at_nat(C,A2) ) ) ).

% sup.strict_boundedE
tff(fact_7303_sup_Ostrict__boundedE,axiom,
    ! [B3: real,C: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,B3),C)),A2))
     => ~ ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
         => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A2)) ) ) ).

% sup.strict_boundedE
tff(fact_7304_sup_Ostrict__boundedE,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B3),C)),A2))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A2)) ) ) ).

% sup.strict_boundedE
tff(fact_7305_sup_Ostrict__boundedE,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B3),C)),A2))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A2)) ) ) ).

% sup.strict_boundedE
tff(fact_7306_sup_Ostrict__boundedE,axiom,
    ! [B3: int,C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B3),C)),A2))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A2)) ) ) ).

% sup.strict_boundedE
tff(fact_7307_sup_Ostrict__order__iff,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B3),A2))
    <=> ( ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% sup.strict_order_iff
tff(fact_7308_sup_Ostrict__order__iff,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ord_le6428140832669894131at_nat(B3,A2)
    <=> ( ( A2 = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% sup.strict_order_iff
tff(fact_7309_sup_Ostrict__order__iff,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(B3,A2)
    <=> ( ( A2 = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% sup.strict_order_iff
tff(fact_7310_sup_Ostrict__order__iff,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
    <=> ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% sup.strict_order_iff
tff(fact_7311_sup_Ostrict__order__iff,axiom,
    ! [B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),A2))
    <=> ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% sup.strict_order_iff
tff(fact_7312_sup_Ostrict__order__iff,axiom,
    ! [B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
    <=> ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% sup.strict_order_iff
tff(fact_7313_sup_Ostrict__order__iff,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2))
    <=> ( ( A2 = aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3) )
        & ( A2 != B3 ) ) ) ).

% sup.strict_order_iff
tff(fact_7314_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_nat,A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,C),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3))) ) ).

% sup.strict_coboundedI1
tff(fact_7315_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ord_le6428140832669894131at_nat(C,A2)
     => ord_le6428140832669894131at_nat(C,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3)) ) ).

% sup.strict_coboundedI1
tff(fact_7316_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(C,A2)
     => ord_le2604355607129572851at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3)) ) ).

% sup.strict_coboundedI1
tff(fact_7317_sup_Ostrict__coboundedI1,axiom,
    ! [C: real,A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A2),B3))) ) ).

% sup.strict_coboundedI1
tff(fact_7318_sup_Ostrict__coboundedI1,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3))) ) ).

% sup.strict_coboundedI1
tff(fact_7319_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3))) ) ).

% sup.strict_coboundedI1
tff(fact_7320_sup_Ostrict__coboundedI1,axiom,
    ! [C: int,A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3))) ) ).

% sup.strict_coboundedI1
tff(fact_7321_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_nat,B3: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,C),B3))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3))) ) ).

% sup.strict_coboundedI2
tff(fact_7322_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ord_le6428140832669894131at_nat(C,B3)
     => ord_le6428140832669894131at_nat(C,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A2),B3)) ) ).

% sup.strict_coboundedI2
tff(fact_7323_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(C,B3)
     => ord_le2604355607129572851at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B3)) ) ).

% sup.strict_coboundedI2
tff(fact_7324_sup_Ostrict__coboundedI2,axiom,
    ! [C: real,B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A2),B3))) ) ).

% sup.strict_coboundedI2
tff(fact_7325_sup_Ostrict__coboundedI2,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A2),B3))) ) ).

% sup.strict_coboundedI2
tff(fact_7326_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B3: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B3))) ) ).

% sup.strict_coboundedI2
tff(fact_7327_sup_Ostrict__coboundedI2,axiom,
    ! [C: int,B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B3))) ) ).

% sup.strict_coboundedI2
tff(fact_7328_distrib__sup__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z)))) ).

% distrib_sup_le
tff(fact_7329_distrib__sup__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)))) ).

% distrib_sup_le
tff(fact_7330_distrib__sup__le,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,Y),Z)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Y)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,X),Z))) ).

% distrib_sup_le
tff(fact_7331_distrib__sup__le,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z))) ).

% distrib_sup_le
tff(fact_7332_distrib__sup__le,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Y),Z))),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Z)))) ).

% distrib_sup_le
tff(fact_7333_distrib__sup__le,axiom,
    ! [X: rat,Y: rat,Z: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z))),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Z)))) ).

% distrib_sup_le
tff(fact_7334_distrib__sup__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z)))) ).

% distrib_sup_le
tff(fact_7335_distrib__sup__le,axiom,
    ! [X: int,Y: int,Z: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Z)))) ).

% distrib_sup_le
tff(fact_7336_distrib__inf__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)))) ).

% distrib_inf_le
tff(fact_7337_distrib__inf__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)))) ).

% distrib_inf_le
tff(fact_7338_distrib__inf__le,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : ord_le3000389064537975527at_nat(aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),Y)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),Z)),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,X),aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_7339_distrib__inf__le,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Z)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_7340_distrib__inf__le,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Z))),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Y),Z)))) ).

% distrib_inf_le
tff(fact_7341_distrib__inf__le,axiom,
    ! [X: rat,Y: rat,Z: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Z))),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Y),Z)))) ).

% distrib_inf_le
tff(fact_7342_distrib__inf__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)))) ).

% distrib_inf_le
tff(fact_7343_distrib__inf__le,axiom,
    ! [X: int,Y: int,Z: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)))) ).

% distrib_inf_le
tff(fact_7344_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K4: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_7345_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K4: real,N: nat] :
      ( ( H != zero_zero_complex )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z),H))),K4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),N)),aa_nat_complex(power_power_complex(Z),N))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_7346_set__encode__insert,axiom,
    ! [A3: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ~ pp(aa_set_nat_bool(member_nat(N),A3))
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(N,A3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_set_nat_nat(nat_set_encode,A3)) ) ) ) ).

% set_encode_insert
tff(fact_7347_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys2: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_int(Ys2))))
     => ( nth_Pr4439495888332055232nt_int(product_int_int(Xs,Ys2),N) = product_Pair_int_int(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_int(Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_7348_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys2: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_s6755466524823107622T_VEBT(Ys2))))
     => ( nth_Pr3474266648193625910T_VEBT(produc662631939642741121T_VEBT(Xs,Ys2),N) = produc3329399203697025711T_VEBT(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys2))),nth_VEBT_VEBT(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys2)))) ) ) ).

% product_nth
tff(fact_7349_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys2: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_bool(Ys2))))
     => ( nth_Pr6495399312670224281t_bool(product_int_bool(Xs,Ys2),N) = produc788913908665201810t_bool(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_bool(Ys2))),aa_nat_bool(nth_bool(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_bool(Ys2)))) ) ) ).

% product_nth
tff(fact_7350_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys2: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_nat(Ys2))))
     => ( nth_Pr8617346907841251940nt_nat(product_int_nat(Xs,Ys2),N) = product_Pair_int_nat(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_nat(Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_nat(Ys2)))) ) ) ).

% product_nth
tff(fact_7351_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys2))))
     => ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys2),N) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_int(Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_7352_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys2))))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys2),N) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys2))),nth_VEBT_VEBT(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys2)))) ) ) ).

% product_nth
tff(fact_7353_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_bool(Ys2))))
     => ( nth_Pr1289831921885633937T_bool(produc5466675335756156456T_bool(Xs,Ys2),N) = produc3035007008238225530T_bool(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_bool(Ys2))),aa_nat_bool(nth_bool(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_bool(Ys2)))) ) ) ).

% product_nth
tff(fact_7354_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys2))))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys2),N) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_nat(Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_nat(Ys2)))) ) ) ).

% product_nth
tff(fact_7355_product__nth,axiom,
    ! [N: nat,Xs: list_bool,Ys2: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_int(Ys2))))
     => ( nth_Pr7231943226834047147ol_int(product_bool_int(Xs,Ys2),N) = produc8871513391732212460ol_int(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_int(Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_7356_product__nth,axiom,
    ! [N: nat,Xs: list_bool,Ys2: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_s6755466524823107622T_VEBT(Ys2))))
     => ( nth_Pr3250499561091541003T_VEBT(produc2282853104913622818T_VEBT(Xs,Ys2),N) = produc9074556814250467700T_VEBT(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys2))),nth_VEBT_VEBT(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys2)))) ) ) ).

% product_nth
tff(fact_7357_unset__bit__0,axiom,
    ! [A2: code_integer] : bit_se8260200283734997820nteger(zero_zero_nat,A2) = times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),divide6298287555418463151nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_7358_unset__bit__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_7359_unset__bit__0,axiom,
    ! [A2: int] : aa_int_int(bit_se4203085406695923979it_int(zero_zero_nat),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_7360_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
         => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) ) ) ) ).

% succ_less_length_list
tff(fact_7361_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X))
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% succ_greatereq_min
tff(fact_7362_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect_nat(vEBT_vebt_member(T)) ).

% set_vebt'_def
tff(fact_7363_finite__Collect__conjI,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( ( pp(aa_set_real_bool(finite_finite_real,collect_real(P)))
        | pp(aa_set_real_bool(finite_finite_real,collect_real(Q))) )
     => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_af(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_7364_finite__Collect__conjI,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ( finite8100373058378681591st_nat(collect_list_nat(P))
        | finite8100373058378681591st_nat(collect_list_nat(Q)) )
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_ag(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_7365_finite__Collect__conjI,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P)))
        | pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(Q))) )
     => pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_ah(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_7366_finite__Collect__conjI,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
        | pp(aa_set_nat_bool(finite_finite_nat,collect_nat(Q))) )
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_ai(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_7367_finite__Collect__conjI,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ( pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
        | pp(aa_set_int_bool(finite_finite_int,collect_int(Q))) )
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_aj(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_7368_finite__Collect__conjI,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P)))
        | pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(Q))) )
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_ak(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_7369_finite__Collect__conjI,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,Q: fun_Pr7757163693017731481t_bool] :
      ( ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P)))
        | pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Q))) )
     => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun830836278467416944t_bool(aTP_Lamm_al(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_7370_finite__Collect__conjI,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,Q: fun_Ex8563670519792538868t_bool] :
      ( ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(P)))
        | pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(Q))) )
     => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_am(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_7371_finite__Collect__disjI,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_an(P),Q))))
    <=> ( pp(aa_set_real_bool(finite_finite_real,collect_real(P)))
        & pp(aa_set_real_bool(finite_finite_real,collect_real(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_7372_finite__Collect__disjI,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( finite8100373058378681591st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_ao(P),Q)))
    <=> ( finite8100373058378681591st_nat(collect_list_nat(P))
        & finite8100373058378681591st_nat(collect_list_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_7373_finite__Collect__disjI,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_ap(P),Q))))
    <=> ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P)))
        & pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_7374_finite__Collect__disjI,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_aq(P),Q))))
    <=> ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
        & pp(aa_set_nat_bool(finite_finite_nat,collect_nat(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_7375_finite__Collect__disjI,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_ar(P),Q))))
    <=> ( pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
        & pp(aa_set_int_bool(finite_finite_int,collect_int(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_7376_finite__Collect__disjI,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_as(P),Q))))
    <=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P)))
        & pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_7377_finite__Collect__disjI,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,Q: fun_Pr7757163693017731481t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun830836278467416944t_bool(aTP_Lamm_at(P),Q))))
    <=> ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P)))
        & pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_7378_finite__Collect__disjI,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,Q: fun_Ex8563670519792538868t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_au(P),Q))))
    <=> ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(P)))
        & pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_7379_succ__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_succ(T,X) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_av(T),X)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_7380_pred__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_pred(T,X) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_aw(T),X)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_7381_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se4203085406695923979it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_7382_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se4203085406695923979it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% unset_bit_negative_int_iff
tff(fact_7383_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_ax(N),C)))) ) ).

% finite_nth_roots
tff(fact_7384_finite__Collect__subsets,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_ay,A3)))) ) ).

% finite_Collect_subsets
tff(fact_7385_finite__Collect__subsets,axiom,
    ! [A3: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_az(A3))) ) ).

% finite_Collect_subsets
tff(fact_7386_finite__Collect__subsets,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => finite9047747110432174090at_nat(collec5514110066124741708at_nat(aTP_Lamm_ba(A3))) ) ).

% finite_Collect_subsets
tff(fact_7387_finite__Collect__subsets,axiom,
    ! [A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => finite5468666774076196335d_enat(collec2260605976452661553d_enat(aTP_Lamm_bb(A3))) ) ).

% finite_Collect_subsets
tff(fact_7388_finite__Collect__subsets,axiom,
    ! [A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_bc(A3))) ) ).

% finite_Collect_subsets
tff(fact_7389_singleton__conv2,axiom,
    ! [A2: product_prod_nat_nat] : collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(fequal9188845342289841878at_nat,A2)) = insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat) ).

% singleton_conv2
tff(fact_7390_singleton__conv2,axiom,
    ! [A2: list_nat] : collect_list_nat(aa_lis4768068104879709938t_bool(fequal_list_nat,A2)) = insert_list_nat(A2,bot_bot_set_list_nat) ).

% singleton_conv2
tff(fact_7391_singleton__conv2,axiom,
    ! [A2: set_nat] : collect_set_nat(aa_set5116768664143533222t_bool(fequal_set_nat,A2)) = insert_set_nat(A2,bot_bot_set_set_nat) ).

% singleton_conv2
tff(fact_7392_singleton__conv2,axiom,
    ! [A2: real] : collect_real(aa_rea7809428808904475482l_bool(fequal_real,A2)) = insert_real(A2,bot_bot_set_real) ).

% singleton_conv2
tff(fact_7393_singleton__conv2,axiom,
    ! [A2: bool] : collect_bool(aa_boo8970175072276269406l_bool(fequal_bool,A2)) = insert_bool(A2,bot_bot_set_bool) ).

% singleton_conv2
tff(fact_7394_singleton__conv2,axiom,
    ! [A2: nat] : collect_nat(aa_nat_fun_nat_bool(fequal_nat,A2)) = insert_nat(A2,bot_bot_set_nat) ).

% singleton_conv2
tff(fact_7395_singleton__conv2,axiom,
    ! [A2: int] : collect_int(aa_int_fun_int_bool(fequal_int,A2)) = insert_int(A2,bot_bot_set_int) ).

% singleton_conv2
tff(fact_7396_singleton__conv,axiom,
    ! [A2: product_prod_nat_nat] : collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(aTP_Lamm_bd,A2)) = insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat) ).

% singleton_conv
tff(fact_7397_singleton__conv,axiom,
    ! [A2: list_nat] : collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_be,A2)) = insert_list_nat(A2,bot_bot_set_list_nat) ).

% singleton_conv
tff(fact_7398_singleton__conv,axiom,
    ! [A2: set_nat] : collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_bf,A2)) = insert_set_nat(A2,bot_bot_set_set_nat) ).

% singleton_conv
tff(fact_7399_singleton__conv,axiom,
    ! [A2: real] : collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_bg,A2)) = insert_real(A2,bot_bot_set_real) ).

% singleton_conv
tff(fact_7400_singleton__conv,axiom,
    ! [A2: bool] : collect_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_bh,A2)) = insert_bool(A2,bot_bot_set_bool) ).

% singleton_conv
tff(fact_7401_singleton__conv,axiom,
    ! [A2: nat] : collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bi,A2)) = insert_nat(A2,bot_bot_set_nat) ).

% singleton_conv
tff(fact_7402_singleton__conv,axiom,
    ! [A2: int] : collect_int(aa_int_fun_int_bool(aTP_Lamm_bj,A2)) = insert_int(A2,bot_bot_set_int) ).

% singleton_conv
tff(fact_7403_finite__Collect__less__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bk,K)))) ).

% finite_Collect_less_nat
tff(fact_7404_finite__Collect__le__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bl,K)))) ).

% finite_Collect_le_nat
tff(fact_7405_card__Collect__less__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bk,N))) = N ).

% card_Collect_less_nat
tff(fact_7406_finite__interval__int1,axiom,
    ! [A2: int,B3: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_bm(A2),B3)))) ).

% finite_interval_int1
tff(fact_7407_finite__interval__int4,axiom,
    ! [A2: int,B3: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_bn(A2),B3)))) ).

% finite_interval_int4
tff(fact_7408_norm__eq__zero,axiom,
    ! [X: real] :
      ( ( aa_real_real(real_V7735802525324610683m_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% norm_eq_zero
tff(fact_7409_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,X) = zero_zero_real )
    <=> ( X = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_7410_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_7411_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_7412_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_7413_card__Collect__le__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bl,N))) = aa_nat_nat(suc,N) ).

% card_Collect_le_nat
tff(fact_7414_finite__interval__int2,axiom,
    ! [A2: int,B3: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_bo(A2),B3)))) ).

% finite_interval_int2
tff(fact_7415_finite__interval__int3,axiom,
    ! [A2: int,B3: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_bp(A2),B3)))) ).

% finite_interval_int3
tff(fact_7416_zero__less__norm__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,X)))
    <=> ( X != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_7417_zero__less__norm__iff,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X)))
    <=> ( X != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_7418_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X)),zero_zero_real))
    <=> ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_7419_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real))
    <=> ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_7420_del__x__not__mia,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
               => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none_nat)),Mi),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),vEBT_vebt_delete(Summary,H)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),H))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),Summary)) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_7421_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L) )
               => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                 => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList2,H,Newnode) )
                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(Sn)),none_nat)),Mi),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(Sn)))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,aa_option_nat_nat(the_nat,vEBT_vebt_maxt(Sn)))))))),Ma))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_7422_del__x__not__mi,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L) )
               => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList2,H,Newnode) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                   => ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none_nat)),Mi),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,aa_option_nat_nat(the_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))))))),Ma))),Deg,Newlist,vEBT_vebt_delete(Summary,H)) ) )
                      & ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_7423_del__in__range,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = 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nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(th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t(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)) ) ) ) ) ).

% del_in_range
tff(fact_7424_del__x__mi,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                 => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none_nat)),Xn),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),vEBT_vebt_delete(Summary,H)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),H))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),Summary)) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_7425_del__x__mi__lets__in,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList2,H,Newnode) )
                     => ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none_nat)),Xn),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,aa_option_nat_nat(the_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))))))),Ma))),Deg,Newlist,vEBT_vebt_delete(Summary,H)) ) )
                        & ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_7426_del__x__mi__lets__in__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList2,H,Newnode) )
                     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(Sn)),none_nat)),Xn),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(Sn)))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,aa_option_nat_nat(the_nat,vEBT_vebt_maxt(Sn)))))))),Ma))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_7427_del__x__mia,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = if_VEBT_VEBT(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)) ) ) ) ) ).

% del_x_mia
tff(fact_7428_pred__less__length__list,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
         => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X),some_nat(Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) ) ) ) ).

% pred_less_length_list
tff(fact_7429_pred__lesseq__max,axiom,
    ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma))
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X),some_nat(Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% pred_lesseq_max
tff(fact_7430_max__def__raw,axiom,
    ! [X4: set_int,Xa: set_int] :
      ( ( pp(aa_set_int_bool(ord_less_eq_set_int(X4),Xa))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,X4),Xa) = Xa ) )
      & ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(X4),Xa))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,X4),Xa) = X4 ) ) ) ).

% max_def_raw
tff(fact_7431_max__def__raw,axiom,
    ! [X4: rat,Xa: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X4),Xa) = Xa ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X4),Xa) = X4 ) ) ) ).

% max_def_raw
tff(fact_7432_max__def__raw,axiom,
    ! [X4: num,Xa: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Xa))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X4),Xa) = Xa ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Xa))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X4),Xa) = X4 ) ) ) ).

% max_def_raw
tff(fact_7433_max__def__raw,axiom,
    ! [X4: nat,Xa: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X4),Xa) = Xa ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X4),Xa) = X4 ) ) ) ).

% max_def_raw
tff(fact_7434_max__def__raw,axiom,
    ! [X4: int,Xa: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X4),Xa) = Xa ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X4),Xa) = X4 ) ) ) ).

% max_def_raw
tff(fact_7435_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_bool,I: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bq(P),I)))) ).

% finite_M_bounded_by_nat
tff(fact_7436_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_br(C),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_7437_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_bs,N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_7438_minus__set__def,axiom,
    ! [A3: set_bool,B2: set_bool] : minus_minus_set_bool(A3,B2) = collect_bool(minus_4822186092517116124l_bool(aTP_Lamm_a(A3),aTP_Lamm_a(B2))) ).

% minus_set_def
tff(fact_7439_minus__set__def,axiom,
    ! [A3: set_real,B2: set_real] : minus_minus_set_real(A3,B2) = collect_real(minus_6582619742941512235l_bool(aTP_Lamm_aa(A3),aTP_Lamm_aa(B2))) ).

% minus_set_def
tff(fact_7440_minus__set__def,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] : minus_7954133019191499631st_nat(A3,B2) = collect_list_nat(minus_8064998606961450743t_bool(aTP_Lamm_ab(A3),aTP_Lamm_ab(B2))) ).

% minus_set_def
tff(fact_7441_minus__set__def,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] : minus_2163939370556025621et_nat(A3,B2) = collect_set_nat(minus_8322047804725131473t_bool(aTP_Lamm_ac(A3),aTP_Lamm_ac(B2))) ).

% minus_set_def
tff(fact_7442_minus__set__def,axiom,
    ! [A3: set_int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2) = collect_int(minus_7727386867957376939t_bool(aTP_Lamm_ae(A3),aTP_Lamm_ae(B2))) ).

% minus_set_def
tff(fact_7443_minus__set__def,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2) = collect_nat(minus_1628305797571225351t_bool(aTP_Lamm_ad(A3),aTP_Lamm_ad(B2))) ).

% minus_set_def
tff(fact_7444_set__diff__eq,axiom,
    ! [A3: set_bool,B2: set_bool] : minus_minus_set_bool(A3,B2) = collect_bool(aa_set6670506438128111678l_bool(aTP_Lamm_bt(A3),B2)) ).

% set_diff_eq
tff(fact_7445_set__diff__eq,axiom,
    ! [A3: set_real,B2: set_real] : minus_minus_set_real(A3,B2) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_bu(A3),B2)) ).

% set_diff_eq
tff(fact_7446_set__diff__eq,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] : minus_7954133019191499631st_nat(A3,B2) = collect_list_nat(aa_set257547205302447420t_bool(aTP_Lamm_bv(A3),B2)) ).

% set_diff_eq
tff(fact_7447_set__diff__eq,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] : minus_2163939370556025621et_nat(A3,B2) = collect_set_nat(aa_set9121396911425987952t_bool(aTP_Lamm_bw(A3),B2)) ).

% set_diff_eq
tff(fact_7448_set__diff__eq,axiom,
    ! [A3: set_int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_bx(A3),B2)) ).

% set_diff_eq
tff(fact_7449_set__diff__eq,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_by(A3),B2)) ).

% set_diff_eq
tff(fact_7450_lambda__zero,axiom,
    aTP_Lamm_bz = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_7451_lambda__zero,axiom,
    aTP_Lamm_ca = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_7452_lambda__zero,axiom,
    aTP_Lamm_cb = aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat) ).

% lambda_zero
tff(fact_7453_lambda__zero,axiom,
    aTP_Lamm_cc = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_7454_lambda__zero,axiom,
    aTP_Lamm_cd = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_7455_less__set__def,axiom,
    ! [A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A3),B2))
    <=> ord_le7519368973566061808l_bool(aTP_Lamm_aa(A3),aTP_Lamm_aa(B2)) ) ).

% less_set_def
tff(fact_7456_less__set__def,axiom,
    ! [A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_set_bool(A3),B2))
    <=> ord_le2077783555311705303l_bool(aTP_Lamm_a(A3),aTP_Lamm_a(B2)) ) ).

% less_set_def
tff(fact_7457_less__set__def,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A3),B2))
    <=> ord_le4523237222650602006t_bool(aTP_Lamm_ac(A3),aTP_Lamm_ac(B2)) ) ).

% less_set_def
tff(fact_7458_less__set__def,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A3),B2))
    <=> ord_le6841235830588351564t_bool(aTP_Lamm_ad(A3),aTP_Lamm_ad(B2)) ) ).

% less_set_def
tff(fact_7459_less__set__def,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A3),B2))
    <=> ord_le3716944864119727344t_bool(aTP_Lamm_ae(A3),aTP_Lamm_ae(B2)) ) ).

% less_set_def
tff(fact_7460_Un__def,axiom,
    ! [A3: set_bool,B2: set_bool] : sup_sup_set_bool(A3,B2) = collect_bool(aa_set6670506438128111678l_bool(aTP_Lamm_ce(A3),B2)) ).

% Un_def
tff(fact_7461_Un__def,axiom,
    ! [A3: set_real,B2: set_real] : sup_sup_set_real(A3,B2) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_cf(A3),B2)) ).

% Un_def
tff(fact_7462_Un__def,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] : sup_sup_set_list_nat(A3,B2) = collect_list_nat(aa_set257547205302447420t_bool(aTP_Lamm_cg(A3),B2)) ).

% Un_def
tff(fact_7463_Un__def,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] : sup_sup_set_set_nat(A3,B2) = collect_set_nat(aa_set9121396911425987952t_bool(aTP_Lamm_ch(A3),B2)) ).

% Un_def
tff(fact_7464_Un__def,axiom,
    ! [A3: set_int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_ci(A3),B2)) ).

% Un_def
tff(fact_7465_Un__def,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_cj(A3),B2)) ).

% Un_def
tff(fact_7466_Un__def,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2) = collec7088162979684241874at_nat(aa_set6997699984206591038t_bool(aTP_Lamm_ck(A3),B2)) ).

% Un_def
tff(fact_7467_Un__def,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2) = collec6321179662152712658at_nat(aa_set377272637606474814t_bool(aTP_Lamm_cl(A3),B2)) ).

% Un_def
tff(fact_7468_sup__set__def,axiom,
    ! [A3: set_bool,B2: set_bool] : sup_sup_set_bool(A3,B2) = collect_bool(sup_su2675241097686107055l_bool(aTP_Lamm_a(A3),aTP_Lamm_a(B2))) ).

% sup_set_def
tff(fact_7469_sup__set__def,axiom,
    ! [A3: set_real,B2: set_real] : sup_sup_set_real(A3,B2) = collect_real(sup_su805534559149407256l_bool(aTP_Lamm_aa(A3),aTP_Lamm_aa(B2))) ).

% sup_set_def
tff(fact_7470_sup__set__def,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] : sup_sup_set_list_nat(A3,B2) = collect_list_nat(sup_su7820324534880642276t_bool(aTP_Lamm_ab(A3),aTP_Lamm_ab(B2))) ).

% sup_set_def
tff(fact_7471_sup__set__def,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] : sup_sup_set_set_nat(A3,B2) = collect_set_nat(sup_su7038862894554899774t_bool(aTP_Lamm_ac(A3),aTP_Lamm_ac(B2))) ).

% sup_set_def
tff(fact_7472_sup__set__def,axiom,
    ! [A3: set_int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2) = collect_int(sup_sup_fun_int_bool(aTP_Lamm_ae(A3),aTP_Lamm_ae(B2))) ).

% sup_set_def
tff(fact_7473_sup__set__def,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2) = collect_nat(sup_sup_fun_nat_bool(aTP_Lamm_ad(A3),aTP_Lamm_ad(B2))) ).

% sup_set_def
tff(fact_7474_sup__set__def,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2) = collec7088162979684241874at_nat(sup_su3901408610078580575t_bool(aTP_Lamm_cm(A3),aTP_Lamm_cm(B2))) ).

% sup_set_def
tff(fact_7475_sup__set__def,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2) = collec6321179662152712658at_nat(sup_su5834520091132235615t_bool(aTP_Lamm_cn(A3),aTP_Lamm_cn(B2))) ).

% sup_set_def
tff(fact_7476_Collect__disj__eq,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] : collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_an(P),Q)) = sup_sup_set_real(collect_real(P),collect_real(Q)) ).

% Collect_disj_eq
tff(fact_7477_Collect__disj__eq,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] : collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_ao(P),Q)) = sup_sup_set_list_nat(collect_list_nat(P),collect_list_nat(Q)) ).

% Collect_disj_eq
tff(fact_7478_Collect__disj__eq,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] : collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_ap(P),Q)) = sup_sup_set_set_nat(collect_set_nat(P),collect_set_nat(Q)) ).

% Collect_disj_eq
tff(fact_7479_Collect__disj__eq,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] : collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_ar(P),Q)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,collect_int(P)),collect_int(Q)) ).

% Collect_disj_eq
tff(fact_7480_Collect__disj__eq,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] : collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_aq(P),Q)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,collect_nat(P)),collect_nat(Q)) ).

% Collect_disj_eq
tff(fact_7481_Collect__disj__eq,axiom,
    ! [P: fun_Pr446604468264085555t_bool,Q: fun_Pr446604468264085555t_bool] : collec7088162979684241874at_nat(aa_fun3665199231186260388t_bool(aTP_Lamm_co(P),Q)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,collec7088162979684241874at_nat(P)),collec7088162979684241874at_nat(Q)) ).

% Collect_disj_eq
tff(fact_7482_Collect__disj__eq,axiom,
    ! [P: fun_Pr1607443439640623155t_bool,Q: fun_Pr1607443439640623155t_bool] : collec6321179662152712658at_nat(aa_fun5022828243006652836t_bool(aTP_Lamm_cp(P),Q)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,collec6321179662152712658at_nat(P)),collec6321179662152712658at_nat(Q)) ).

% Collect_disj_eq
tff(fact_7483_insert__def,axiom,
    ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A2,B2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(aTP_Lamm_bd,A2))),B2) ).

% insert_def
tff(fact_7484_insert__def,axiom,
    ! [A2: bool,B2: set_bool] : insert_bool(A2,B2) = sup_sup_set_bool(collect_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_bh,A2)),B2) ).

% insert_def
tff(fact_7485_insert__def,axiom,
    ! [A2: real,B2: set_real] : insert_real(A2,B2) = sup_sup_set_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_bg,A2)),B2) ).

% insert_def
tff(fact_7486_insert__def,axiom,
    ! [A2: list_nat,B2: set_list_nat] : insert_list_nat(A2,B2) = sup_sup_set_list_nat(collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_be,A2)),B2) ).

% insert_def
tff(fact_7487_insert__def,axiom,
    ! [A2: set_nat,B2: set_set_nat] : insert_set_nat(A2,B2) = sup_sup_set_set_nat(collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_bf,A2)),B2) ).

% insert_def
tff(fact_7488_insert__def,axiom,
    ! [A2: int,B2: set_int] : insert_int(A2,B2) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_bj,A2))),B2) ).

% insert_def
tff(fact_7489_insert__def,axiom,
    ! [A2: nat,B2: set_nat] : insert_nat(A2,B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bi,A2))),B2) ).

% insert_def
tff(fact_7490_insert__def,axiom,
    ! [A2: produc859450856879609959at_nat,B2: set_Pr8693737435421807431at_nat] : insert5050368324300391991at_nat(A2,B2) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,collec7088162979684241874at_nat(aTP_Lamm_cq(A2))),B2) ).

% insert_def
tff(fact_7491_insert__def,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A2,B2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,collec6321179662152712658at_nat(aTP_Lamm_cr(A2))),B2) ).

% insert_def
tff(fact_7492_Compl__eq,axiom,
    ! [A3: set_bool] : uminus5166143097222041326t_bool(A3) = collect_bool(aTP_Lamm_cs(A3)) ).

% Compl_eq
tff(fact_7493_Compl__eq,axiom,
    ! [A3: set_real] : uminus612125837232591019t_real(A3) = collect_real(aTP_Lamm_ct(A3)) ).

% Compl_eq
tff(fact_7494_Compl__eq,axiom,
    ! [A3: set_list_nat] : uminus3195874150345416415st_nat(A3) = collect_list_nat(aTP_Lamm_cu(A3)) ).

% Compl_eq
tff(fact_7495_Compl__eq,axiom,
    ! [A3: set_set_nat] : uminus613421341184616069et_nat(A3) = collect_set_nat(aTP_Lamm_cv(A3)) ).

% Compl_eq
tff(fact_7496_Compl__eq,axiom,
    ! [A3: set_nat] : uminus5710092332889474511et_nat(A3) = collect_nat(aTP_Lamm_cw(A3)) ).

% Compl_eq
tff(fact_7497_Compl__eq,axiom,
    ! [A3: set_int] : uminus1532241313380277803et_int(A3) = collect_int(aTP_Lamm_cx(A3)) ).

% Compl_eq
tff(fact_7498_Collect__neg__eq,axiom,
    ! [P: fun_real_bool] : collect_real(aTP_Lamm_cy(P)) = uminus612125837232591019t_real(collect_real(P)) ).

% Collect_neg_eq
tff(fact_7499_Collect__neg__eq,axiom,
    ! [P: fun_list_nat_bool] : collect_list_nat(aTP_Lamm_cz(P)) = uminus3195874150345416415st_nat(collect_list_nat(P)) ).

% Collect_neg_eq
tff(fact_7500_Collect__neg__eq,axiom,
    ! [P: fun_set_nat_bool] : collect_set_nat(aTP_Lamm_da(P)) = uminus613421341184616069et_nat(collect_set_nat(P)) ).

% Collect_neg_eq
tff(fact_7501_Collect__neg__eq,axiom,
    ! [P: fun_nat_bool] : collect_nat(aTP_Lamm_db(P)) = uminus5710092332889474511et_nat(collect_nat(P)) ).

% Collect_neg_eq
tff(fact_7502_Collect__neg__eq,axiom,
    ! [P: fun_int_bool] : collect_int(aTP_Lamm_dc(P)) = uminus1532241313380277803et_int(collect_int(P)) ).

% Collect_neg_eq
tff(fact_7503_uminus__set__def,axiom,
    ! [A3: set_bool] : uminus5166143097222041326t_bool(A3) = collect_bool(uminus215040010513443916l_bool(aTP_Lamm_a(A3))) ).

% uminus_set_def
tff(fact_7504_uminus__set__def,axiom,
    ! [A3: set_real] : uminus612125837232591019t_real(A3) = collect_real(uminus4638318004487240379l_bool(aTP_Lamm_aa(A3))) ).

% uminus_set_def
tff(fact_7505_uminus__set__def,axiom,
    ! [A3: set_list_nat] : uminus3195874150345416415st_nat(A3) = collect_list_nat(uminus5119773305880778631t_bool(aTP_Lamm_ab(A3))) ).

% uminus_set_def
tff(fact_7506_uminus__set__def,axiom,
    ! [A3: set_set_nat] : uminus613421341184616069et_nat(A3) = collect_set_nat(uminus6226608070181861729t_bool(aTP_Lamm_ac(A3))) ).

% uminus_set_def
tff(fact_7507_uminus__set__def,axiom,
    ! [A3: set_nat] : uminus5710092332889474511et_nat(A3) = collect_nat(uminus1995910650241132439t_bool(aTP_Lamm_ad(A3))) ).

% uminus_set_def
tff(fact_7508_uminus__set__def,axiom,
    ! [A3: set_int] : uminus1532241313380277803et_int(A3) = collect_int(uminus8094991720627284027t_bool(aTP_Lamm_ae(A3))) ).

% uminus_set_def
tff(fact_7509_not__finite__existsD,axiom,
    ! [P: fun_real_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,collect_real(P)))
     => ? [X_12: real] : pp(aa_real_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_7510_not__finite__existsD,axiom,
    ! [P: fun_list_nat_bool] :
      ( ~ finite8100373058378681591st_nat(collect_list_nat(P))
     => ? [X_12: list_nat] : pp(aa_list_nat_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_7511_not__finite__existsD,axiom,
    ! [P: fun_set_nat_bool] :
      ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P)))
     => ? [X_12: set_nat] : pp(aa_set_nat_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_7512_not__finite__existsD,axiom,
    ! [P: fun_nat_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => ? [X_12: nat] : pp(aa_nat_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_7513_not__finite__existsD,axiom,
    ! [P: fun_int_bool] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
     => ? [X_12: int] : pp(aa_int_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_7514_not__finite__existsD,axiom,
    ! [P: fun_complex_bool] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P)))
     => ? [X_12: complex] : pp(aa_complex_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_7515_not__finite__existsD,axiom,
    ! [P: fun_Pr7757163693017731481t_bool] :
      ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P)))
     => ? [X_12: product_prod_nat_nat] : pp(aa_Pro2232559814647311250t_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_7516_not__finite__existsD,axiom,
    ! [P: fun_Ex8563670519792538868t_bool] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(P)))
     => ? [X_12: extended_enat] : pp(aa_Ext2772979874139974381t_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_7517_pigeonhole__infinite__rel,axiom,
    ! [A3: set_bool,B2: set_nat,R: fun_bo3288154511880838028t_bool] :
      ( ~ pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ! [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),A3))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B2))
                  & pp(aa_nat_bool(aa_bool_fun_nat_bool(R,X5),Xa)) ) )
         => ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),B2))
              & ~ pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_nat_fun_bool_bool(aa_fun4163026448321506586l_bool(aTP_Lamm_dd(A3),R),X5)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7518_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B2: set_nat,R: fun_re9121486324101951821t_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),A3))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B2))
                  & pp(aa_nat_bool(aa_real_fun_nat_bool(R,X5),Xa)) ) )
         => ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),B2))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_de(A3),R),X5)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7519_pigeonhole__infinite__rel,axiom,
    ! [A3: set_bool,B2: set_int,R: fun_bo2827871560234191408t_bool] :
      ( ~ pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( ! [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),A3))
             => ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),B2))
                  & pp(aa_int_bool(aa_bool_fun_int_bool(R,X5),Xa)) ) )
         => ? [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),B2))
              & ~ pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_int_fun_bool_bool(aa_fun6035399485891414554l_bool(aTP_Lamm_df(A3),R),X5)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7520_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B2: set_int,R: fun_re8661203372455305201t_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),A3))
             => ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),B2))
                  & pp(aa_int_bool(aa_real_fun_int_bool(R,X5),Xa)) ) )
         => ? [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),B2))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_dg(A3),R),X5)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7521_pigeonhole__infinite__rel,axiom,
    ! [A3: set_bool,B2: set_complex,R: fun_bo8977660800532043054x_bool] :
      ( ~ pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ! [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),A3))
             => ? [Xa: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa),B2))
                  & pp(aa_complex_bool(aa_boo8509281623719168167x_bool(R,X5),Xa)) ) )
         => ? [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),B2))
              & ~ pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_com4332222969190720879l_bool(aa_fun6951469949174074266l_bool(aTP_Lamm_dh(A3),R),X5)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7522_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B2: set_complex,R: fun_re3010880487650781423x_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),A3))
             => ? [Xa: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa),B2))
                  & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(R,X5),Xa)) ) )
         => ? [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),B2))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_di(A3),R),X5)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7523_pigeonhole__infinite__rel,axiom,
    ! [A3: set_bool,B2: set_Extended_enat,R: fun_bo6623066019567229854t_bool] :
      ( ~ pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( ! [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),A3))
             => ? [Xa: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),B2))
                  & pp(aa_Ext2772979874139974381t_bool(aa_boo1364755444788961487t_bool(R,X5),Xa)) ) )
         => ? [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),B2))
              & ~ pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_Ext5859091071952444205l_bool(aa_fun4004702039914900634l_bool(aTP_Lamm_dj(A3),R),X5)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7524_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B2: set_Extended_enat,R: fun_re7357811994743134493t_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),A3))
             => ? [Xa: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),B2))
                  & pp(aa_Ext2772979874139974381t_bool(aa_rea3571252896447469150t_bool(R,X5),Xa)) ) )
         => ? [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),B2))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_Ext4143593917056713754l_bool(aa_fun3666023268991729404l_bool(aTP_Lamm_dk(A3),R),X5)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7525_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B2: set_nat,R: fun_nat_fun_nat_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ! [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),A3))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B2))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X5),Xa)) ) )
         => ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),B2))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_dl(A3),R),X5)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7526_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B2: set_int,R: fun_nat_fun_int_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( ! [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),A3))
             => ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),B2))
                  & pp(aa_int_bool(aa_nat_fun_int_bool(R,X5),Xa)) ) )
         => ? [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),B2))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_dm(A3),R),X5)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_7527_Collect__conv__if,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,A2: product_prod_nat_nat] :
      ( ( pp(aa_Pro2232559814647311250t_bool(P,A2))
       => ( collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(aTP_Lamm_dn(P),A2)) = insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat) ) )
      & ( ~ pp(aa_Pro2232559814647311250t_bool(P,A2))
       => ( collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(aTP_Lamm_dn(P),A2)) = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if
tff(fact_7528_Collect__conv__if,axiom,
    ! [P: fun_list_nat_bool,A2: list_nat] :
      ( ( pp(aa_list_nat_bool(P,A2))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_do(P),A2)) = insert_list_nat(A2,bot_bot_set_list_nat) ) )
      & ( ~ pp(aa_list_nat_bool(P,A2))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_do(P),A2)) = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if
tff(fact_7529_Collect__conv__if,axiom,
    ! [P: fun_set_nat_bool,A2: set_nat] :
      ( ( pp(aa_set_nat_bool(P,A2))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_dp(P),A2)) = insert_set_nat(A2,bot_bot_set_set_nat) ) )
      & ( ~ pp(aa_set_nat_bool(P,A2))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_dp(P),A2)) = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if
tff(fact_7530_Collect__conv__if,axiom,
    ! [P: fun_real_bool,A2: real] :
      ( ( pp(aa_real_bool(P,A2))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_dq(P),A2)) = insert_real(A2,bot_bot_set_real) ) )
      & ( ~ pp(aa_real_bool(P,A2))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_dq(P),A2)) = bot_bot_set_real ) ) ) ).

% Collect_conv_if
tff(fact_7531_Collect__conv__if,axiom,
    ! [P: fun_bool_bool,A2: bool] :
      ( ( pp(aa_bool_bool(P,A2))
       => ( collect_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_dr(P),A2)) = insert_bool(A2,bot_bot_set_bool) ) )
      & ( ~ pp(aa_bool_bool(P,A2))
       => ( collect_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_dr(P),A2)) = bot_bot_set_bool ) ) ) ).

% Collect_conv_if
tff(fact_7532_Collect__conv__if,axiom,
    ! [P: fun_nat_bool,A2: nat] :
      ( ( pp(aa_nat_bool(P,A2))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ds(P),A2)) = insert_nat(A2,bot_bot_set_nat) ) )
      & ( ~ pp(aa_nat_bool(P,A2))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ds(P),A2)) = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
tff(fact_7533_Collect__conv__if,axiom,
    ! [P: fun_int_bool,A2: int] :
      ( ( pp(aa_int_bool(P,A2))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_dt(P),A2)) = insert_int(A2,bot_bot_set_int) ) )
      & ( ~ pp(aa_int_bool(P,A2))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_dt(P),A2)) = bot_bot_set_int ) ) ) ).

% Collect_conv_if
tff(fact_7534_Collect__conv__if2,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,A2: product_prod_nat_nat] :
      ( ( pp(aa_Pro2232559814647311250t_bool(P,A2))
       => ( collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(aTP_Lamm_du(P),A2)) = insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat) ) )
      & ( ~ pp(aa_Pro2232559814647311250t_bool(P,A2))
       => ( collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(aTP_Lamm_du(P),A2)) = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if2
tff(fact_7535_Collect__conv__if2,axiom,
    ! [P: fun_list_nat_bool,A2: list_nat] :
      ( ( pp(aa_list_nat_bool(P,A2))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_dv(P),A2)) = insert_list_nat(A2,bot_bot_set_list_nat) ) )
      & ( ~ pp(aa_list_nat_bool(P,A2))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_dv(P),A2)) = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if2
tff(fact_7536_Collect__conv__if2,axiom,
    ! [P: fun_set_nat_bool,A2: set_nat] :
      ( ( pp(aa_set_nat_bool(P,A2))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_dw(P),A2)) = insert_set_nat(A2,bot_bot_set_set_nat) ) )
      & ( ~ pp(aa_set_nat_bool(P,A2))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_dw(P),A2)) = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if2
tff(fact_7537_Collect__conv__if2,axiom,
    ! [P: fun_real_bool,A2: real] :
      ( ( pp(aa_real_bool(P,A2))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_dx(P),A2)) = insert_real(A2,bot_bot_set_real) ) )
      & ( ~ pp(aa_real_bool(P,A2))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_dx(P),A2)) = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
tff(fact_7538_Collect__conv__if2,axiom,
    ! [P: fun_bool_bool,A2: bool] :
      ( ( pp(aa_bool_bool(P,A2))
       => ( collect_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_dy(P),A2)) = insert_bool(A2,bot_bot_set_bool) ) )
      & ( ~ pp(aa_bool_bool(P,A2))
       => ( collect_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_dy(P),A2)) = bot_bot_set_bool ) ) ) ).

% Collect_conv_if2
tff(fact_7539_Collect__conv__if2,axiom,
    ! [P: fun_nat_bool,A2: nat] :
      ( ( pp(aa_nat_bool(P,A2))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_dz(P),A2)) = insert_nat(A2,bot_bot_set_nat) ) )
      & ( ~ pp(aa_nat_bool(P,A2))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_dz(P),A2)) = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
tff(fact_7540_Collect__conv__if2,axiom,
    ! [P: fun_int_bool,A2: int] :
      ( ( pp(aa_int_bool(P,A2))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_ea(P),A2)) = insert_int(A2,bot_bot_set_int) ) )
      & ( ~ pp(aa_int_bool(P,A2))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_ea(P),A2)) = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
tff(fact_7541_insert__Collect,axiom,
    ! [A2: product_prod_nat_nat,P: fun_Pr7757163693017731481t_bool] : insert8211810215607154385at_nat(A2,collec3392354462482085612at_nat(P)) = collec3392354462482085612at_nat(aa_fun830836278467416944t_bool(aTP_Lamm_eb(A2),P)) ).

% insert_Collect
tff(fact_7542_insert__Collect,axiom,
    ! [A2: bool,P: fun_bool_bool] : insert_bool(A2,collect_bool(P)) = collect_bool(aa_fun8098540714523660868l_bool(aTP_Lamm_ec(A2),P)) ).

% insert_Collect
tff(fact_7543_insert__Collect,axiom,
    ! [A2: real,P: fun_real_bool] : insert_real(A2,collect_real(P)) = collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_ed(A2),P)) ).

% insert_Collect
tff(fact_7544_insert__Collect,axiom,
    ! [A2: list_nat,P: fun_list_nat_bool] : insert_list_nat(A2,collect_list_nat(P)) = collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_ee(A2),P)) ).

% insert_Collect
tff(fact_7545_insert__Collect,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] : insert_set_nat(A2,collect_set_nat(P)) = collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_ef(A2),P)) ).

% insert_Collect
tff(fact_7546_insert__Collect,axiom,
    ! [A2: nat,P: fun_nat_bool] : insert_nat(A2,collect_nat(P)) = collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_eg(A2),P)) ).

% insert_Collect
tff(fact_7547_insert__Collect,axiom,
    ! [A2: int,P: fun_int_bool] : insert_int(A2,collect_int(P)) = collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_eh(A2),P)) ).

% insert_Collect
tff(fact_7548_insert__compr,axiom,
    ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A2,B2) = collec3392354462482085612at_nat(aa_set5223363028666124222t_bool(aTP_Lamm_ei(A2),B2)) ).

% insert_compr
tff(fact_7549_insert__compr,axiom,
    ! [A2: bool,B2: set_bool] : insert_bool(A2,B2) = collect_bool(aa_set6670506438128111678l_bool(aTP_Lamm_ej(A2),B2)) ).

% insert_compr
tff(fact_7550_insert__compr,axiom,
    ! [A2: real,B2: set_real] : insert_real(A2,B2) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_ek(A2),B2)) ).

% insert_compr
tff(fact_7551_insert__compr,axiom,
    ! [A2: list_nat,B2: set_list_nat] : insert_list_nat(A2,B2) = collect_list_nat(aa_set257547205302447420t_bool(aTP_Lamm_el(A2),B2)) ).

% insert_compr
tff(fact_7552_insert__compr,axiom,
    ! [A2: set_nat,B2: set_set_nat] : insert_set_nat(A2,B2) = collect_set_nat(aa_set9121396911425987952t_bool(aTP_Lamm_em(A2),B2)) ).

% insert_compr
tff(fact_7553_insert__compr,axiom,
    ! [A2: nat,B2: set_nat] : insert_nat(A2,B2) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_en(A2),B2)) ).

% insert_compr
tff(fact_7554_insert__compr,axiom,
    ! [A2: int,B2: set_int] : insert_int(A2,B2) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_eo(A2),B2)) ).

% insert_compr
tff(fact_7555_empty__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(aTP_Lamm_ep) ).

% empty_def
tff(fact_7556_empty__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(aTP_Lamm_eq) ).

% empty_def
tff(fact_7557_empty__def,axiom,
    bot_bot_set_real = collect_real(aTP_Lamm_er) ).

% empty_def
tff(fact_7558_empty__def,axiom,
    bot_bot_set_bool = collect_bool(aTP_Lamm_es) ).

% empty_def
tff(fact_7559_empty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_et) ).

% empty_def
tff(fact_7560_empty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_eu) ).

% empty_def
tff(fact_7561_Collect__imp__eq,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] : collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_ev(P),Q)) = sup_sup_set_real(uminus612125837232591019t_real(collect_real(P)),collect_real(Q)) ).

% Collect_imp_eq
tff(fact_7562_Collect__imp__eq,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] : collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_ew(P),Q)) = sup_sup_set_list_nat(uminus3195874150345416415st_nat(collect_list_nat(P)),collect_list_nat(Q)) ).

% Collect_imp_eq
tff(fact_7563_Collect__imp__eq,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] : collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_ex(P),Q)) = sup_sup_set_set_nat(uminus613421341184616069et_nat(collect_set_nat(P)),collect_set_nat(Q)) ).

% Collect_imp_eq
tff(fact_7564_Collect__imp__eq,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] : collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_ey(P),Q)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,uminus1532241313380277803et_int(collect_int(P))),collect_int(Q)) ).

% Collect_imp_eq
tff(fact_7565_Collect__imp__eq,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] : collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_ez(P),Q)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(collect_nat(P))),collect_nat(Q)) ).

% Collect_imp_eq
tff(fact_7566_Collect__imp__eq,axiom,
    ! [P: fun_Pr446604468264085555t_bool,Q: fun_Pr446604468264085555t_bool] : collec7088162979684241874at_nat(aa_fun3665199231186260388t_bool(aTP_Lamm_fa(P),Q)) = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,uminus4384627049435823934at_nat(collec7088162979684241874at_nat(P))),collec7088162979684241874at_nat(Q)) ).

% Collect_imp_eq
tff(fact_7567_Collect__imp__eq,axiom,
    ! [P: fun_Pr1607443439640623155t_bool,Q: fun_Pr1607443439640623155t_bool] : collec6321179662152712658at_nat(aa_fun5022828243006652836t_bool(aTP_Lamm_fb(P),Q)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(collec6321179662152712658at_nat(P))),collec6321179662152712658at_nat(Q)) ).

% Collect_imp_eq
tff(fact_7568_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(F,N2)))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_fc(F),U)))) ) ).

% finite_less_ub
tff(fact_7569_bot__empty__eq2,axiom,
    ! [X4: option_nat,Xa: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(bot_bo6687431919080314997t_bool,X4),Xa))
    <=> member4117937158525611210on_nat(produc5098337634421038937on_nat(X4,Xa),bot_bo232370072503712749on_nat) ) ).

% bot_empty_eq2
tff(fact_7570_bot__empty__eq2,axiom,
    ! [X4: set_Pr4329608150637261639at_nat,Xa: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(bot_bo4620695493499315481t_bool,X4),Xa))
    <=> member1466754251312161552at_nat(produc9060074326276436823at_nat(X4,Xa),bot_bo4948859079157340979at_nat) ) ).

% bot_empty_eq2
tff(fact_7571_bot__empty__eq2,axiom,
    ! [X4: set_Pr1261947904930325089at_nat,Xa: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(bot_bo3728952843091432729t_bool,X4),Xa))
    <=> member8757157785044589968at_nat(produc2922128104949294807at_nat(X4,Xa),bot_bo228742789529271731at_nat) ) ).

% bot_empty_eq2
tff(fact_7572_bot__empty__eq2,axiom,
    ! [X4: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bot_bo6560894463593755925t_bool,X4),Xa))
    <=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Xa)),bot_bo2099793752762293965at_nat)) ) ).

% bot_empty_eq2
tff(fact_7573_bot__empty__eq2,axiom,
    ! [X4: int,Xa: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(bot_bo6207810721336151133t_bool,X4),Xa))
    <=> member5262025264175285858nt_int(product_Pair_int_int(X4,Xa),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_7574_pred__subset__eq,axiom,
    ! [R: set_real,S: set_real] :
      ( ord_le5573776112191778788l_bool(aTP_Lamm_aa(R),aTP_Lamm_aa(S))
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(R),S)) ) ).

% pred_subset_eq
tff(fact_7575_pred__subset__eq,axiom,
    ! [R: set_bool,S: set_bool] :
      ( ord_le2902245940885495011l_bool(aTP_Lamm_a(R),aTP_Lamm_a(S))
    <=> pp(aa_set_bool_bool(ord_less_eq_set_bool(R),S)) ) ).

% pred_subset_eq
tff(fact_7576_pred__subset__eq,axiom,
    ! [R: set_set_nat,S: set_set_nat] :
      ( ord_le1461730252015488778t_bool(aTP_Lamm_ac(R),aTP_Lamm_ac(S))
    <=> pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(R),S)) ) ).

% pred_subset_eq
tff(fact_7577_pred__subset__eq,axiom,
    ! [R: set_nat,S: set_nat] :
      ( ord_le8840051076831633728t_bool(aTP_Lamm_ad(R),aTP_Lamm_ad(S))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,R),S)) ) ).

% pred_subset_eq
tff(fact_7578_pred__subset__eq,axiom,
    ! [R: set_int,S: set_int] :
      ( ord_le5715760110363009508t_bool(aTP_Lamm_ae(R),aTP_Lamm_ae(S))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(R),S)) ) ).

% pred_subset_eq
tff(fact_7579_prop__restrict,axiom,
    ! [X: bool,Z5: set_bool,X6: set_bool,P: fun_bool_bool] :
      ( pp(aa_set_bool_bool(member_bool(X),Z5))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(Z5),collect_bool(aa_fun8098540714523660868l_bool(aTP_Lamm_fd(X6),P))))
       => pp(aa_bool_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_7580_prop__restrict,axiom,
    ! [X: real,Z5: set_real,X6: set_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(member_real(X),Z5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(Z5),collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_fe(X6),P))))
       => pp(aa_real_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_7581_prop__restrict,axiom,
    ! [X: list_nat,Z5: set_list_nat,X6: set_list_nat,P: fun_list_nat_bool] :
      ( member_list_nat(X,Z5)
     => ( ord_le6045566169113846134st_nat(Z5,collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_ff(X6),P)))
       => pp(aa_list_nat_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_7582_prop__restrict,axiom,
    ! [X: set_nat,Z5: set_set_nat,X6: set_set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),Z5))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(Z5),collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_fg(X6),P))))
       => pp(aa_set_nat_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_7583_prop__restrict,axiom,
    ! [X: nat,Z5: set_nat,X6: set_nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(member_nat(X),Z5))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Z5),collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_fh(X6),P))))
       => pp(aa_nat_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_7584_prop__restrict,axiom,
    ! [X: int,Z5: set_int,X6: set_int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(member_int(X),Z5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Z5),collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_fi(X6),P))))
       => pp(aa_int_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_7585_Collect__restrict,axiom,
    ! [X6: set_bool,P: fun_bool_bool] : pp(aa_set_bool_bool(ord_less_eq_set_bool(collect_bool(aa_fun8098540714523660868l_bool(aTP_Lamm_fd(X6),P))),X6)) ).

% Collect_restrict
tff(fact_7586_Collect__restrict,axiom,
    ! [X6: set_real,P: fun_real_bool] : pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_fe(X6),P))),X6)) ).

% Collect_restrict
tff(fact_7587_Collect__restrict,axiom,
    ! [X6: set_list_nat,P: fun_list_nat_bool] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_ff(X6),P)),X6) ).

% Collect_restrict
tff(fact_7588_Collect__restrict,axiom,
    ! [X6: set_set_nat,P: fun_set_nat_bool] : pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_fg(X6),P))),X6)) ).

% Collect_restrict
tff(fact_7589_Collect__restrict,axiom,
    ! [X6: set_nat,P: fun_nat_bool] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_fh(X6),P))),X6)) ).

% Collect_restrict
tff(fact_7590_Collect__restrict,axiom,
    ! [X6: set_int,P: fun_int_bool] : pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_fi(X6),P))),X6)) ).

% Collect_restrict
tff(fact_7591_less__eq__set__def,axiom,
    ! [A3: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2))
    <=> ord_le5573776112191778788l_bool(aTP_Lamm_aa(A3),aTP_Lamm_aa(B2)) ) ).

% less_eq_set_def
tff(fact_7592_less__eq__set__def,axiom,
    ! [A3: set_bool,B2: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),B2))
    <=> ord_le2902245940885495011l_bool(aTP_Lamm_a(A3),aTP_Lamm_a(B2)) ) ).

% less_eq_set_def
tff(fact_7593_less__eq__set__def,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A3),B2))
    <=> ord_le1461730252015488778t_bool(aTP_Lamm_ac(A3),aTP_Lamm_ac(B2)) ) ).

% less_eq_set_def
tff(fact_7594_less__eq__set__def,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
    <=> ord_le8840051076831633728t_bool(aTP_Lamm_ad(A3),aTP_Lamm_ad(B2)) ) ).

% less_eq_set_def
tff(fact_7595_less__eq__set__def,axiom,
    ! [A3: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B2))
    <=> ord_le5715760110363009508t_bool(aTP_Lamm_ae(A3),aTP_Lamm_ae(B2)) ) ).

% less_eq_set_def
tff(fact_7596_Collect__subset,axiom,
    ! [A3: set_bool,P: fun_bool_bool] : pp(aa_set_bool_bool(ord_less_eq_set_bool(collect_bool(aa_fun8098540714523660868l_bool(aTP_Lamm_fd(A3),P))),A3)) ).

% Collect_subset
tff(fact_7597_Collect__subset,axiom,
    ! [A3: set_real,P: fun_real_bool] : pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_fe(A3),P))),A3)) ).

% Collect_subset
tff(fact_7598_Collect__subset,axiom,
    ! [A3: set_list_nat,P: fun_list_nat_bool] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_ff(A3),P)),A3) ).

% Collect_subset
tff(fact_7599_Collect__subset,axiom,
    ! [A3: set_set_nat,P: fun_set_nat_bool] : pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_fg(A3),P))),A3)) ).

% Collect_subset
tff(fact_7600_Collect__subset,axiom,
    ! [A3: set_nat,P: fun_nat_bool] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_fh(A3),P))),A3)) ).

% Collect_subset
tff(fact_7601_Collect__subset,axiom,
    ! [A3: set_int,P: fun_int_bool] : pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_fi(A3),P))),A3)) ).

% Collect_subset
tff(fact_7602_pred__subset__eq2,axiom,
    ! [R: set_Pr6588086440996610945on_nat,S: set_Pr6588086440996610945on_nat] :
      ( ord_le7152797810718812585t_bool(aTP_Lamm_fj(R),aTP_Lamm_fj(S))
    <=> ord_le6406482658798684961on_nat(R,S) ) ).

% pred_subset_eq2
tff(fact_7603_pred__subset__eq2,axiom,
    ! [R: set_Pr7459493094073627847at_nat,S: set_Pr7459493094073627847at_nat] :
      ( ord_le3287501503639976013t_bool(aTP_Lamm_fk(R),aTP_Lamm_fk(S))
    <=> ord_le5997549366648089703at_nat(R,S) ) ).

% pred_subset_eq2
tff(fact_7604_pred__subset__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ord_le1428200584844665933t_bool(aTP_Lamm_fl(R),aTP_Lamm_fl(S))
    <=> ord_le1268244103169919719at_nat(R,S) ) ).

% pred_subset_eq2
tff(fact_7605_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ord_le4899916226251588169t_bool(aTP_Lamm_fm(R),aTP_Lamm_fm(S))
    <=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(R),S)) ) ).

% pred_subset_eq2
tff(fact_7606_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ord_le4546832483993983377t_bool(aTP_Lamm_fn(R),aTP_Lamm_fn(S))
    <=> ord_le2843351958646193337nt_int(R,S) ) ).

% pred_subset_eq2
tff(fact_7607_Int__def,axiom,
    ! [A3: set_bool,B2: set_bool] : inf_inf_set_bool(A3,B2) = collect_bool(aa_set6670506438128111678l_bool(aTP_Lamm_fo(A3),B2)) ).

% Int_def
tff(fact_7608_Int__def,axiom,
    ! [A3: set_real,B2: set_real] : inf_inf_set_real(A3,B2) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_fp(A3),B2)) ).

% Int_def
tff(fact_7609_Int__def,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] : inf_inf_set_list_nat(A3,B2) = collect_list_nat(aa_set257547205302447420t_bool(aTP_Lamm_fq(A3),B2)) ).

% Int_def
tff(fact_7610_Int__def,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] : inf_inf_set_set_nat(A3,B2) = collect_set_nat(aa_set9121396911425987952t_bool(aTP_Lamm_fr(A3),B2)) ).

% Int_def
tff(fact_7611_Int__def,axiom,
    ! [A3: set_int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_fs(A3),B2)) ).

% Int_def
tff(fact_7612_Int__def,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_ft(A3),B2)) ).

% Int_def
tff(fact_7613_Int__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) = collec3392354462482085612at_nat(aa_set5223363028666124222t_bool(aTP_Lamm_fu(A3),B2)) ).

% Int_def
tff(fact_7614_Int__Collect,axiom,
    ! [X: bool,A3: set_bool,P: fun_bool_bool] :
      ( pp(aa_set_bool_bool(member_bool(X),inf_inf_set_bool(A3,collect_bool(P))))
    <=> ( pp(aa_set_bool_bool(member_bool(X),A3))
        & pp(aa_bool_bool(P,X)) ) ) ).

% Int_Collect
tff(fact_7615_Int__Collect,axiom,
    ! [X: real,A3: set_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(member_real(X),inf_inf_set_real(A3,collect_real(P))))
    <=> ( pp(aa_set_real_bool(member_real(X),A3))
        & pp(aa_real_bool(P,X)) ) ) ).

% Int_Collect
tff(fact_7616_Int__Collect,axiom,
    ! [X: list_nat,A3: set_list_nat,P: fun_list_nat_bool] :
      ( member_list_nat(X,inf_inf_set_list_nat(A3,collect_list_nat(P)))
    <=> ( member_list_nat(X,A3)
        & pp(aa_list_nat_bool(P,X)) ) ) ).

% Int_Collect
tff(fact_7617_Int__Collect,axiom,
    ! [X: set_nat,A3: set_set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),inf_inf_set_set_nat(A3,collect_set_nat(P))))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(X),A3))
        & pp(aa_set_nat_bool(P,X)) ) ) ).

% Int_Collect
tff(fact_7618_Int__Collect,axiom,
    ! [X: int,A3: set_int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(P))))
    <=> ( pp(aa_set_int_bool(member_int(X),A3))
        & pp(aa_int_bool(P,X)) ) ) ).

% Int_Collect
tff(fact_7619_Int__Collect,axiom,
    ! [X: nat,A3: set_nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),collect_nat(P))))
    <=> ( pp(aa_set_nat_bool(member_nat(X),A3))
        & pp(aa_nat_bool(P,X)) ) ) ).

% Int_Collect
tff(fact_7620_Int__Collect,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,P: fun_Pr7757163693017731481t_bool] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),collec3392354462482085612at_nat(P))))
    <=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A3))
        & pp(aa_Pro2232559814647311250t_bool(P,X)) ) ) ).

% Int_Collect
tff(fact_7621_inf__set__def,axiom,
    ! [A3: set_bool,B2: set_bool] : inf_inf_set_bool(A3,B2) = collect_bool(inf_in9217375510005350933l_bool(aTP_Lamm_a(A3),aTP_Lamm_a(B2))) ).

% inf_set_def
tff(fact_7622_inf__set__def,axiom,
    ! [A3: set_real,B2: set_real] : inf_inf_set_real(A3,B2) = collect_real(inf_in4003345473364989746l_bool(aTP_Lamm_aa(A3),aTP_Lamm_aa(B2))) ).

% inf_set_def
tff(fact_7623_inf__set__def,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] : inf_inf_set_list_nat(A3,B2) = collect_list_nat(inf_in7940471347505502206t_bool(aTP_Lamm_ab(A3),aTP_Lamm_ab(B2))) ).

% inf_set_def
tff(fact_7624_inf__set__def,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] : inf_inf_set_set_nat(A3,B2) = collect_set_nat(inf_in6168126700042863960t_bool(aTP_Lamm_ac(A3),aTP_Lamm_ac(B2))) ).

% inf_set_def
tff(fact_7625_inf__set__def,axiom,
    ! [A3: set_int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2) = collect_int(inf_inf_fun_int_bool(aTP_Lamm_ae(A3),aTP_Lamm_ae(B2))) ).

% inf_set_def
tff(fact_7626_inf__set__def,axiom,
    ! [A3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2) = collect_nat(inf_inf_fun_nat_bool(aTP_Lamm_ad(A3),aTP_Lamm_ad(B2))) ).

% inf_set_def
tff(fact_7627_inf__set__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2) = collec3392354462482085612at_nat(inf_in2244499300375407147t_bool(aTP_Lamm_fv(A3),aTP_Lamm_fv(B2))) ).

% inf_set_def
tff(fact_7628_Collect__conj__eq,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] : collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_af(P),Q)) = inf_inf_set_real(collect_real(P),collect_real(Q)) ).

% Collect_conj_eq
tff(fact_7629_Collect__conj__eq,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] : collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_ag(P),Q)) = inf_inf_set_list_nat(collect_list_nat(P),collect_list_nat(Q)) ).

% Collect_conj_eq
tff(fact_7630_Collect__conj__eq,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] : collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_ah(P),Q)) = inf_inf_set_set_nat(collect_set_nat(P),collect_set_nat(Q)) ).

% Collect_conj_eq
tff(fact_7631_Collect__conj__eq,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] : collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_aj(P),Q)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,collect_int(P)),collect_int(Q)) ).

% Collect_conj_eq
tff(fact_7632_Collect__conj__eq,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] : collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_ai(P),Q)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,collect_nat(P)),collect_nat(Q)) ).

% Collect_conj_eq
tff(fact_7633_Collect__conj__eq,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,Q: fun_Pr7757163693017731481t_bool] : collec3392354462482085612at_nat(aa_fun830836278467416944t_bool(aTP_Lamm_al(P),Q)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,collec3392354462482085612at_nat(P)),collec3392354462482085612at_nat(Q)) ).

% Collect_conj_eq
tff(fact_7634_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T)) ).

% set_vebt_def
tff(fact_7635_finite__int__segment,axiom,
    ! [A2: real,B3: real] : pp(aa_set_real_bool(finite_finite_real,collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_fw(A2),B3)))) ).

% finite_int_segment
tff(fact_7636_finite__int__segment,axiom,
    ! [A2: rat,B3: rat] : pp(aa_set_rat_bool(finite_finite_rat,collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_fx(A2),B3)))) ).

% finite_int_segment
tff(fact_7637_nat__less__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_less_as_int
tff(fact_7638_nat__leq__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_leq_as_int
tff(fact_7639_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se4203085406695923979it_int(N),K)),K)) ).

% unset_bit_less_eq
tff(fact_7640_finite__abs__int__segment,axiom,
    ! [A2: real] : pp(aa_set_real_bool(finite_finite_real,collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_fy,A2)))) ).

% finite_abs_int_segment
tff(fact_7641_finite__abs__int__segment,axiom,
    ! [A2: rat] : pp(aa_set_rat_bool(finite_finite_rat,collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_fz,A2)))) ).

% finite_abs_int_segment
tff(fact_7642_card__less,axiom,
    ! [M5: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M5))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ga(M5),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_7643_card__less__Suc,axiom,
    ! [M5: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M5))
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gb(M5),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ga(M5),I))) ) ) ).

% card_less_Suc
tff(fact_7644_card__less__Suc2,axiom,
    ! [M5: set_nat,I: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M5))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gb(M5),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ga(M5),I))) ) ) ).

% card_less_Suc2
tff(fact_7645_norm__not__less__zero,axiom,
    ! [X: complex] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real)) ).

% norm_not_less_zero
tff(fact_7646_norm__ge__zero,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X))) ).

% norm_ge_zero
tff(fact_7647_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,X))),aa_complex_real(real_V1022390504157884413omplex,X))) ).

% complex_mod_minus_le_complex_mod
tff(fact_7648_complex__mod__triangle__ineq2,axiom,
    ! [B3: complex,A2: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(B3),A2))),aa_complex_real(real_V1022390504157884413omplex,B3))),aa_complex_real(real_V1022390504157884413omplex,A2))) ).

% complex_mod_triangle_ineq2
tff(fact_7649_set__encode__eq,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ( aa_set_nat_nat(nat_set_encode,A3) = aa_set_nat_nat(nat_set_encode,B2) )
        <=> ( A3 = B2 ) ) ) ) ).

% set_encode_eq
tff(fact_7650_finite__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_gc,N)))) ) ).

% finite_roots_unity
tff(fact_7651_finite__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_bs,N)))) ) ).

% finite_roots_unity
tff(fact_7652_card__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(collect_real(aa_nat_fun_real_bool(aTP_Lamm_gc,N)))),N)) ) ).

% card_roots_unity
tff(fact_7653_card__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_bs,N)))),N)) ) ).

% card_roots_unity
tff(fact_7654_finite__lists__length__eq,axiom,
    ! [A3: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_gd(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_7655_finite__lists__length__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1772366067986465313t_bool(aTP_Lamm_ge(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_7656_finite__lists__length__eq,axiom,
    ! [A3: set_Extended_enat,N: nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => finite1862508098717546133d_enat(collec8433460942617342167d_enat(aa_nat4017767907309459068t_bool(aTP_Lamm_gf(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_7657_finite__lists__length__eq,axiom,
    ! [A3: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A3)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_gg(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_7658_finite__lists__length__eq,axiom,
    ! [A3: set_bool,N: nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_gh(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_7659_finite__lists__length__eq,axiom,
    ! [A3: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_gi(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_7660_finite__lists__length__eq,axiom,
    ! [A3: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_gj(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_7661_card__lists__length__eq,axiom,
    ! [A3: set_list_nat,N: nat] :
      ( finite8100373058378681591st_nat(A3)
     => ( finite7325466520557071688st_nat(collec5989764272469232197st_nat(aa_nat3401988846471794802t_bool(aTP_Lamm_gk(A3),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_list_nat(A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_7662_card__lists__length__eq,axiom,
    ! [A3: set_set_nat,N: nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A3))
     => ( finite5631907774883551598et_nat(collect_list_set_nat(aa_nat948039967059030092t_bool(aTP_Lamm_gl(A3),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_set_nat(A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_7663_card__lists__length__eq,axiom,
    ! [A3: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( finite5120063068150530198omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_gd(A3),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_complex(A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_7664_card__lists__length__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( finite249151656366948015at_nat(collec3343600615725829874at_nat(aa_nat1772366067986465313t_bool(aTP_Lamm_ge(A3),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_7665_card__lists__length__eq,axiom,
    ! [A3: set_Extended_enat,N: nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( finite7441382602597825044d_enat(collec8433460942617342167d_enat(aa_nat4017767907309459068t_bool(aTP_Lamm_gf(A3),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite121521170596916366d_enat(A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_7666_card__lists__length__eq,axiom,
    ! [A3: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A3)
     => ( finite5915292604075114978T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_gg(A3),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite7802652506058667612T_VEBT(A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_7667_card__lists__length__eq,axiom,
    ! [A3: set_bool,N: nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( finite1699927739564962373t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_gh(A3),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_bool(A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_7668_card__lists__length__eq,axiom,
    ! [A3: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_gi(A3),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set_nat_nat(finite_card_nat,A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_7669_card__lists__length__eq,axiom,
    ! [A3: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( finite_card_list_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_gj(A3),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_int(A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_7670_finite__lists__length__le,axiom,
    ! [A3: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_gm(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7671_finite__lists__length__le,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1772366067986465313t_bool(aTP_Lamm_gn(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7672_finite__lists__length__le,axiom,
    ! [A3: set_Extended_enat,N: nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => finite1862508098717546133d_enat(collec8433460942617342167d_enat(aa_nat4017767907309459068t_bool(aTP_Lamm_go(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7673_finite__lists__length__le,axiom,
    ! [A3: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A3)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_gp(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7674_finite__lists__length__le,axiom,
    ! [A3: set_bool,N: nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_gq(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7675_finite__lists__length__le,axiom,
    ! [A3: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_gr(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7676_finite__lists__length__le,axiom,
    ! [A3: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_gs(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7677_nonzero__norm__divide,axiom,
    ! [B3: real,A2: real] :
      ( ( B3 != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B3)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B3)) ) ) ).

% nonzero_norm_divide
tff(fact_7678_nonzero__norm__divide,axiom,
    ! [B3: complex,A2: complex] :
      ( ( B3 != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A2),B3)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B3)) ) ) ).

% nonzero_norm_divide
tff(fact_7679_power__eq__imp__eq__norm,axiom,
    ! [W2: real,N: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_real_real(real_V7735802525324610683m_real,W2) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_7680_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,N: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W2),N) = aa_nat_complex(power_power_complex(Z),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_complex_real(real_V1022390504157884413omplex,W2) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_7681_norm__mult__less,axiom,
    ! [X: real,R2: real,Y: real,S2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S2))) ) ) ).

% norm_mult_less
tff(fact_7682_norm__mult__less,axiom,
    ! [X: complex,R2: real,Y: complex,S2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S2))) ) ) ).

% norm_mult_less
tff(fact_7683_norm__mult__ineq,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y)))) ).

% norm_mult_ineq
tff(fact_7684_norm__mult__ineq,axiom,
    ! [X: complex,Y: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))) ).

% norm_mult_ineq
tff(fact_7685_norm__add__less,axiom,
    ! [X: real,R2: real,Y: real,S2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S2))) ) ) ).

% norm_add_less
tff(fact_7686_norm__add__less,axiom,
    ! [X: complex,R2: real,Y: complex,S2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S2))) ) ) ).

% norm_add_less
tff(fact_7687_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),E2)) ) ).

% norm_triangle_lt
tff(fact_7688_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E2)) ) ).

% norm_triangle_lt
tff(fact_7689_norm__power__ineq,axiom,
    ! [X: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,X)),N))) ).

% norm_power_ineq
tff(fact_7690_norm__power__ineq,axiom,
    ! [X: complex,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(X),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,X)),N))) ).

% norm_power_ineq
tff(fact_7691_norm__add__leD,axiom,
    ! [A2: real,B3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3))),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B3)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A2)),C))) ) ).

% norm_add_leD
tff(fact_7692_norm__add__leD,axiom,
    ! [A2: complex,B3: complex,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A2),B3))),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B3)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A2)),C))) ) ).

% norm_add_leD
tff(fact_7693_norm__triangle__le,axiom,
    ! [X: real,Y: real,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),E2)) ) ).

% norm_triangle_le
tff(fact_7694_norm__triangle__le,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E2)) ) ).

% norm_triangle_le
tff(fact_7695_norm__triangle__ineq,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y)))) ).

% norm_triangle_ineq
tff(fact_7696_norm__triangle__ineq,axiom,
    ! [X: complex,Y: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))) ).

% norm_triangle_ineq
tff(fact_7697_norm__triangle__mono,axiom,
    ! [A2: real,R2: real,B3: real,S2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A2)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B3)),S2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S2))) ) ) ).

% norm_triangle_mono
tff(fact_7698_norm__triangle__mono,axiom,
    ! [A2: complex,R2: real,B3: complex,S2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A2)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B3)),S2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A2),B3))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S2))) ) ) ).

% norm_triangle_mono
tff(fact_7699_norm__diff__triangle__less,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_less
tff(fact_7700_norm__diff__triangle__less,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_less
tff(fact_7701_norm__triangle__sub,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Y)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))))) ).

% norm_triangle_sub
tff(fact_7702_norm__triangle__sub,axiom,
    ! [X: complex,Y: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Y)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))))) ).

% norm_triangle_sub
tff(fact_7703_norm__triangle__ineq4,axiom,
    ! [A2: real,B3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B3)))) ).

% norm_triangle_ineq4
tff(fact_7704_norm__triangle__ineq4,axiom,
    ! [A2: complex,B3: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A2),B3))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B3)))) ).

% norm_triangle_ineq4
tff(fact_7705_norm__diff__triangle__le,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_le
tff(fact_7706_norm__diff__triangle__le,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_le
tff(fact_7707_norm__triangle__le__diff,axiom,
    ! [X: real,Y: real,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))),E2)) ) ).

% norm_triangle_le_diff
tff(fact_7708_norm__triangle__le__diff,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E2)) ) ).

% norm_triangle_le_diff
tff(fact_7709_norm__diff__ineq,axiom,
    ! [A2: real,B3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B3))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)))) ).

% norm_diff_ineq
tff(fact_7710_norm__diff__ineq,axiom,
    ! [A2: complex,B3: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B3))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A2),B3)))) ).

% norm_diff_ineq
tff(fact_7711_norm__triangle__ineq2,axiom,
    ! [A2: real,B3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B3))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)))) ).

% norm_triangle_ineq2
tff(fact_7712_norm__triangle__ineq2,axiom,
    ! [A2: complex,B3: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B3))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A2),B3)))) ).

% norm_triangle_ineq2
tff(fact_7713_norm__exp,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,X))),aa_real_real(exp_real,aa_real_real(real_V7735802525324610683m_real,X)))) ).

% norm_exp
tff(fact_7714_norm__exp,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,X))),aa_real_real(exp_real,aa_complex_real(real_V1022390504157884413omplex,X)))) ).

% norm_exp
tff(fact_7715_set__encode__inf,axiom,
    ! [A3: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( aa_set_nat_nat(nat_set_encode,A3) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_7716_power__eq__1__iff,axiom,
    ! [W2: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),N) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W2) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_7717_power__eq__1__iff,axiom,
    ! [W2: complex,N: nat] :
      ( ( aa_nat_complex(power_power_complex(W2),N) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W2) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_7718_norm__diff__triangle__ineq,axiom,
    ! [A2: real,B3: real,C: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B3)),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,A2),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),D))))) ).

% norm_diff_triangle_ineq
tff(fact_7719_norm__diff__triangle__ineq,axiom,
    ! [A2: complex,B3: complex,C: complex,D: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A2),B3)),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(A2),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B3),D))))) ).

% norm_diff_triangle_ineq
tff(fact_7720_norm__sgn,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(sgn_sgn_real,X)) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(sgn_sgn_real,X)) = one_one_real ) ) ) ).

% norm_sgn
tff(fact_7721_norm__sgn,axiom,
    ! [X: complex] :
      ( ( ( X = zero_zero_complex )
       => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(sgn_sgn_complex,X)) = zero_zero_real ) )
      & ( ( X != zero_zero_complex )
       => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(sgn_sgn_complex,X)) = one_one_real ) ) ) ).

% norm_sgn
tff(fact_7722_norm__triangle__ineq3,axiom,
    ! [A2: real,B3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B3)))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B3)))) ).

% norm_triangle_ineq3
tff(fact_7723_norm__triangle__ineq3,axiom,
    ! [A2: complex,B3: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B3)))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A2),B3)))) ).

% norm_triangle_ineq3
tff(fact_7724_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList2,S2)),X))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
         => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_7725_norm__power__diff,axiom,
    ! [Z: real,W2: real,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W2)),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(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,W2),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),W2))))) ) ) ).

% norm_power_diff
tff(fact_7726_norm__power__diff,axiom,
    ! [Z: complex,W2: complex,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W2)),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Z),M)),aa_nat_complex(power_power_complex(W2),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),W2))))) ) ) ).

% norm_power_diff
tff(fact_7727_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList2,Vd)),X))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
         => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_7728_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary)),X))
    <=> ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
                 => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                     => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_7729_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList2,Vc)),X))
    <=> ( ( X = Mi )
        | ( X = Ma )
        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_7730_exp__bound__half,axiom,
    ! [Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% exp_bound_half
tff(fact_7731_exp__bound__half,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% exp_bound_half
tff(fact_7732_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
      <=> pp(Y) )
     => ( ! [A: bool,B: bool] :
            ( ( X = vEBT_Leaf(A,B) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero_nat )
                   => pp(A) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
           => pp(Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3)
               => ( pp(Y)
                <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_7733_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( ! [A: bool,B: bool] :
            ( ( X = vEBT_Leaf(A,B) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => pp(A) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => pp(B) )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3)
             => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                   => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_7734_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( ! [A: bool,B: bool] :
            ( ( X = vEBT_Leaf(A,B) )
           => ( ( ( Xa2 = zero_zero_nat )
               => pp(A) )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => pp(B) )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3)
               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                   => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_7735_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2)
             => ~ ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 )
                  | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                     => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) )
         => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2)
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                     => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_7736_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( ! [A: bool,B: bool] :
            ( ( X = vEBT_Leaf(A,B) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => pp(A) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => pp(B) )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2)
             => ~ ( ( Xa2 != Mi2 )
                 => ( ( Xa2 != Ma2 )
                   => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                               => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_7737_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( ! [Uu2: bool,Uv2: bool] : X != vEBT_Leaf(Uu2,Uv2)
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 )
                    | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2)
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_7738_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => pp(Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => pp(Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( pp(Y)
                <=> ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( pp(Y)
                  <=> ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2)
                   => ( pp(Y)
                    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_7739_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary),X) = if_VEBT_VEBT(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary)) ).

% vebt_insert.simps(5)
tff(fact_7740_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
      <=> pp(Y) )
     => ( ! [A: bool,B: bool] :
            ( ( X = vEBT_Leaf(A,B) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero_nat )
                   => pp(A) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => pp(Y) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
             => pp(Y) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => pp(Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2)
                   => ( pp(Y)
                    <=> ~ ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                                       => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_7741_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( ! [A: bool,B: bool] :
            ( ( X = vEBT_Leaf(A,B) )
           => ( ( ( Xa2 = zero_zero_nat )
               => pp(A) )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => pp(B) )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2)
                   => ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                                   => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_7742_exp__bound__lemma,axiom,
    ! [Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(real_V7735802525324610683m_real,Z))))) ) ).

% exp_bound_lemma
tff(fact_7743_exp__bound__lemma,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(real_V1022390504157884413omplex,Z))))) ) ).

% exp_bound_lemma
tff(fact_7744_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y )
     => ( ! [A: bool,B: bool] :
            ( ( X = vEBT_Leaf(A,B) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y = vEBT_Leaf(fTrue,B) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y = vEBT_Leaf(A,fTrue) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y = vEBT_Leaf(A,B) ) ) ) ) ) )
       => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info,zero_zero_nat,Ts,S3) )
             => ( Y != vEBT_Node(Info,zero_zero_nat,Ts,S3) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S3) )
               => ( Y != vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S3) ) )
           => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary2) )
                 => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary2) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2) )
                   => ( Y != if_VEBT_VEBT(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary2)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2)) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_7745_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary),X) = some_nat(Mi) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary),X) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_7746_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary),X) = some_nat(Ma) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary),X) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X),some_nat(Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_7747_vebt__delete_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)) )
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary) ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary),X) = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary),X) = if_VEBT_VEBT(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_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nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary)) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_7748_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa2) = Y )
     => ( ! [A: bool,B: bool] :
            ( ( X = vEBT_Leaf(A,B) )
           => ( ( Xa2 = zero_zero_nat )
             => ( Y != vEBT_Leaf(fFalse,B) ) ) )
       => ( ! [A: bool] :
              ( ? [B: bool] : X = vEBT_Leaf(A,B)
             => ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != vEBT_Leaf(A,fFalse) ) ) )
         => ( ! [A: bool,B: bool] :
                ( ( X = vEBT_Leaf(A,B) )
               => ( ? [N2: nat] : Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,N2))
                 => ( Y != vEBT_Leaf(A,B) ) ) )
           => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2) )
                 => ( Y != vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2,Smry2) )
                   => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2,Smry2) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) )
                     => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2) )
                       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2)) )
                             => ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2) ) )
                            & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2)) )
                             => ( ( ( ( Xa2 = Mi2 )
                                    & ( Xa2 = Ma2 ) )
                                 => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2) ) )
                                & ( ~ ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                 => ( Y = 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8373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary2)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_7749_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(X,Xa2) = Y )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( ( Xa2 = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A: bool] :
              ( ? [Uw2: bool] : X = vEBT_Leaf(A,Uw2)
             => ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
               => ~ ( ( pp(A)
                     => ( Y = some_nat(zero_zero_nat) ) )
                    & ( ~ pp(A)
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A: bool,B: bool] :
                ( ( X = vEBT_Leaf(A,B) )
               => ( ? [Va2: nat] : Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
                 => ~ ( ( pp(B)
                       => ( Y = some_nat(one_one_nat) ) )
                      & ( ~ pp(B)
                       => ( ( pp(A)
                           => ( Y = some_nat(zero_zero_nat) ) )
                          & ( ~ pp(A)
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2)
                 => ( Y != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2)
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2) )
                       => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                             => ( Y = some_nat(Ma2) ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                             => ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa2),some_nat(Mi2),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_7750_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(X,Xa2) = Y )
     => ( ! [Uu2: bool,B: bool] :
            ( ( X = vEBT_Leaf(Uu2,B) )
           => ( ( Xa2 = zero_zero_nat )
             => ~ ( ( pp(B)
                   => ( Y = some_nat(one_one_nat) ) )
                  & ( ~ pp(B)
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv2: bool,Uw2: bool] : X = vEBT_Leaf(Uv2,Uw2)
           => ( ? [N2: nat] : Xa2 = aa_nat_nat(suc,N2)
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2)
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2)
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2) )
                     => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( Y = some_nat(Mi2) ) )
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_7751_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_int(ring_1_of_int_int,K) = zero_zero_int ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(aa_int_fun_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_int)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_7752_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_real(ring_1_of_int_real,K) = zero_zero_real ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_real)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_7753_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_rat(ring_1_of_int_rat,K) = zero_zero_rat ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_rat(ring_1_of_int_rat,K) = aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_rat(ring_1_of_int_rat,K) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_rat)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_7754_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = zero_z3403309356797280102nteger ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_Code_integer)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_7755_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_complex(ring_17405671764205052669omplex,K) = zero_zero_complex ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_complex)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_7756_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(X,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,B: bool] :
              ( ( X = vEBT_Leaf(Uu2,B) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( ( pp(B)
                     => ( Y = some_nat(one_one_nat) ) )
                    & ( ~ pp(B)
                     => ( Y = none_nat ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,B),zero_zero_nat)) ) ) )
         => ( ! [Uv2: bool,Uw2: bool] :
                ( ( X = vEBT_Leaf(Uv2,Uw2) )
               => ! [N2: nat] :
                    ( ( Xa2 = aa_nat_nat(suc,N2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uv2,Uw2),aa_nat_nat(suc,N2))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2) )
                 => ( ( Y = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Xa2)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Xa2)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Xa2)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2) )
                       => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                             => ( Y = some_nat(Mi2) ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                             => ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2),Xa2)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_7757_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(X,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),zero_zero_nat)) ) ) )
         => ( ! [A: bool,Uw2: bool] :
                ( ( X = vEBT_Leaf(A,Uw2) )
               => ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( ( pp(A)
                       => ( Y = some_nat(zero_zero_nat) ) )
                      & ( ~ pp(A)
                       => ( Y = none_nat ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf(A,Uw2),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A: bool,B: bool] :
                  ( ( X = vEBT_Leaf(A,B) )
                 => ! [Va2: nat] :
                      ( ( Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( ( pp(B)
                           => ( Y = some_nat(one_one_nat) ) )
                          & ( ~ pp(B)
                           => ( ( pp(A)
                               => ( Y = some_nat(zero_zero_nat) ) )
                              & ( ~ pp(A)
                               => ( Y = none_nat ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xa2)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2),Xa2)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2) )
                       => ( ( Y = none_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Xa2)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2) )
                         => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                               => ( Y = some_nat(Ma2) ) )
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                               => ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa2),some_nat(Mi2),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_7758_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A: bool,B: bool] :
              ( ( X = vEBT_Leaf(A,B) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = vEBT_Leaf(fFalse,B) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf(A,B),zero_zero_nat)) ) ) )
         => ( ! [A: bool,B: bool] :
                ( ( X = vEBT_Leaf(A,B) )
               => ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = vEBT_Leaf(A,fFalse) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf(A,B),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A: bool,B: bool] :
                  ( ( X = vEBT_Leaf(A,B) )
                 => ! [N2: nat] :
                      ( ( Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                     => ( ( Y = vEBT_Leaf(A,B) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) )
             => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2) )
                   => ( ( Y = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2),Xa2)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2,Smry2) )
                     => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2,Smry2) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2,Smry2),Xa2)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) )
                       => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2),Xa2)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2) )
                         => ( ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2)) )
                               => ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2) ) )
                              & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                    | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2)) )
                               => ( ( ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                   => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2) ) )
                                  & ( ~ ( ( Xa2 = Mi2 )
                                        & ( Xa2 = Ma2 ) )
                                   => ( Y = 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8373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary2)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2)) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_7759_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_gt,X)) ) ).

% monoseq_arctan_series
tff(fact_7760_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A: bool,B: bool] :
              ( ( X = vEBT_Leaf(A,B) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y = vEBT_Leaf(fTrue,B) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y = vEBT_Leaf(A,fTrue) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y = vEBT_Leaf(A,B) ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf(A,B),Xa2)) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info,zero_zero_nat,Ts,S3) )
               => ( ( Y = vEBT_Node(Info,zero_zero_nat,Ts,S3) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info,zero_zero_nat,Ts,S3),Xa2)) ) )
           => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S3) )
                 => ( ( Y = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S3) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S3),Xa2)) ) )
             => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary2) )
                   => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary2),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2) )
                     => ( ( Y = if_VEBT_VEBT(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary2)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_7761_monoseq__realpow,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,X)) ) ) ).

% monoseq_realpow
tff(fact_7762_VEBT__internal_Oinsert_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_insert(X,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_insert_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A: bool,B: bool] :
              ( ( X = vEBT_Leaf(A,B) )
             => ( ( Y = vEBT_vebt_insert(vEBT_Leaf(A,B),Xa2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf(A,B),Xa2)) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
               => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)),Xa2))
                     => ( Y = vEBT_Node(Info,Deg2,TreeList,Summary2) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)),Xa2))
                     => ( Y = vEBT_vebt_insert(vEBT_Node(Info,Deg2,TreeList,Summary2),Xa2) ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info,Deg2,TreeList,Summary2),Xa2)) ) ) ) ) ) ).

% VEBT_internal.insert'.pelims
tff(fact_7763_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A: bool,B: bool] :
              ( ( X = vEBT_Leaf(A,B) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero_nat )
                     => pp(A) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B) )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A,B),Xa2)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa2)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ pp(Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ pp(Y)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2) )
                     => ( ( pp(Y)
                        <=> ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                 => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                                         => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_7764_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A: bool,B: bool] :
              ( ( X = vEBT_Leaf(A,B) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A,B),Xa2))
               => ( ( ( Xa2 = zero_zero_nat )
                   => pp(A) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa2)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2),Xa2))
                       => ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                                       => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_7765_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A: bool,B: bool] :
              ( ( X = vEBT_Leaf(A,B) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A,B),Xa2))
               => ( ( ( Xa2 = zero_zero_nat )
                   => pp(A) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa2)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3),Xa2))
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_7766_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A: bool,B: bool] :
              ( ( X = vEBT_Leaf(A,B) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A,B),Xa2))
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => pp(A) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B) )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3),Xa2))
                 => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_7767_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A: bool,B: bool] :
              ( ( X = vEBT_Leaf(A,B) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero_nat )
                     => pp(A) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B) )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A,B),Xa2)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa2)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3) )
                 => ( ( pp(Y)
                    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                         => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3),Xa2)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_7768_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A: bool,B: bool] :
              ( ( X = vEBT_Leaf(A,B) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A,B),Xa2))
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => pp(A) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B) )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary2),Xa2))
                 => ~ ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                                   => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_7769_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa2)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xa2))
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2),Xa2))
                     => ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),Xa2))
                       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_7770_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ~ pp(Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa2)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
                 => ( ( pp(Y)
                    <=> ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xa2)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2) )
                   => ( ( pp(Y)
                      <=> ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                             => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2),Xa2)) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2) )
                     => ( ( pp(Y)
                        <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                             => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),Xa2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_7771_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xa2))
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2),Xa2))
                 => ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 )
                      | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                         => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) )
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),Xa2))
                   => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                         => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_7772_ln__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(ln_ln_real,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_gu,X)) ) ) ) ).

% ln_series
tff(fact_7773_signed__take__bit__rec,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( ( N = zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N,A2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) )
      & ( ( N != zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N,A2) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide6298287555418463151nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_7774_signed__take__bit__rec,axiom,
    ! [N: nat,A2: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_int_int(bit_ri631733984087533419it_int(N),A2) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
      & ( ( N != zero_zero_nat )
       => ( aa_int_int(bit_ri631733984087533419it_int(N),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_7775_arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(arctan,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_gv,X)) ) ) ).

% arctan_series
tff(fact_7776_signed__take__bit__of__0,axiom,
    ! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(N),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_7777_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_gw(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_7778_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_gx(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_7779_signed__take__bit__0,axiom,
    ! [A2: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_7780_signed__take__bit__0,axiom,
    ! [A2: int] : aa_int_int(bit_ri631733984087533419it_int(zero_zero_nat),A2) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_7781_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% signed_take_bit_int_less_exp
tff(fact_7782_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_ri631733984087533419it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_7783_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K)) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_7784_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_int_int(bit_ri631733984087533419it_int(N),K))) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_7785_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K)) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_7786_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(bit_ri631733984087533419it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_7787_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))))) ) ).

% signed_take_bit_int_less_eq
tff(fact_7788_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(bit_ri631733984087533419it_int(N),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_7789_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(N),K) = K )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_7790_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))),aa_int_int(bit_ri631733984087533419it_int(N),K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_7791_suminf__geometric,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
     => ( suminf_real(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_7792_suminf__geometric,axiom,
    ! [C: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
     => ( suminf_complex(power_power_complex(C)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C)) ) ) ).

% suminf_geometric
tff(fact_7793_suminf__zero,axiom,
    suminf_real(aTP_Lamm_gy) = zero_zero_real ).

% suminf_zero
tff(fact_7794_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_cc) = zero_zero_nat ).

% suminf_zero
tff(fact_7795_suminf__zero,axiom,
    suminf_int(aTP_Lamm_gz) = zero_zero_int ).

% suminf_zero
tff(fact_7796_round__unique,axiom,
    ! [X: real,Y: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Y)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))
       => ( aa_real_int(archim8280529875227126926d_real,X) = Y ) ) ) ).

% round_unique
tff(fact_7797_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = Y ) ) ) ).

% round_unique
tff(fact_7798_summable__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_gv,X)) ) ).

% summable_arctan_series
tff(fact_7799_tanh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_7800_round__unique_H,axiom,
    ! [X: rat,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_int_rat(ring_1_of_int_rat,N)))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
     => ( aa_rat_int(archim7778729529865785530nd_rat,X) = N ) ) ).

% round_unique'
tff(fact_7801_round__unique_H,axiom,
    ! [X: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_int_real(ring_1_of_int_real,N)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_int(archim8280529875227126926d_real,X) = N ) ) ).

% round_unique'
tff(fact_7802_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_7803_tanh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% tanh_real_less_iff
tff(fact_7804_tanh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% tanh_real_le_iff
tff(fact_7805_summable__zero,axiom,
    summable_real(aTP_Lamm_gy) ).

% summable_zero
tff(fact_7806_summable__zero,axiom,
    summable_nat(aTP_Lamm_cc) ).

% summable_zero
tff(fact_7807_summable__zero,axiom,
    summable_int(aTP_Lamm_gz) ).

% summable_zero
tff(fact_7808_summable__single,axiom,
    ! [I: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ha(I),F)) ).

% summable_single
tff(fact_7809_summable__single,axiom,
    ! [I: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hb(I),F)) ).

% summable_single
tff(fact_7810_summable__single,axiom,
    ! [I: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_hc(I),F)) ).

% summable_single
tff(fact_7811_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_7812_round__0,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_7813_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% tanh_real_neg_iff
tff(fact_7814_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(tanh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% tanh_real_pos_iff
tff(fact_7815_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% tanh_real_nonneg_iff
tff(fact_7816_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tanh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% tanh_real_nonpos_iff
tff(fact_7817_summable__cmult__iff,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hd(C),F))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_cmult_iff
tff(fact_7818_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_he(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_7819_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_hf(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_7820_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_hg(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_7821_summable__If__finite__set,axiom,
    ! [A3: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_hh(A3),F)) ) ).

% summable_If_finite_set
tff(fact_7822_summable__If__finite__set,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hi(A3),F)) ) ).

% summable_If_finite_set
tff(fact_7823_summable__If__finite__set,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_hj(A3),F)) ) ).

% summable_If_finite_set
tff(fact_7824_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_hk(P),F)) ) ).

% summable_If_finite
tff(fact_7825_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hl(P),F)) ) ).

% summable_If_finite
tff(fact_7826_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_hm(P),F)) ) ).

% summable_If_finite
tff(fact_7827_summable__geometric__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(power_power_real,C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)) ) ).

% summable_geometric_iff
tff(fact_7828_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( summable_complex(power_power_complex(C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)) ) ).

% summable_geometric_iff
tff(fact_7829_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_hn,C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_7830_summable__comparison__test_H,axiom,
    ! [G3: fun_nat_real,N4: nat,F: fun_nat_real] :
      ( summable_real(G3)
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N4),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))),aa_nat_real(G3,N2))) )
       => summable_real(F) ) ) ).

% summable_comparison_test'
tff(fact_7831_summable__comparison__test_H,axiom,
    ! [G3: fun_nat_real,N4: nat,F: fun_nat_complex] :
      ( summable_real(G3)
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N4),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G3,N2))) )
       => summable_complex(F) ) ) ).

% summable_comparison_test'
tff(fact_7832_summable__comparison__test,axiom,
    ! [F: fun_nat_real,G3: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))),aa_nat_real(G3,N2))) )
     => ( summable_real(G3)
       => summable_real(F) ) ) ).

% summable_comparison_test
tff(fact_7833_summable__comparison__test,axiom,
    ! [F: fun_nat_complex,G3: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G3,N2))) )
     => ( summable_real(G3)
       => summable_complex(F) ) ) ).

% summable_comparison_test
tff(fact_7834_suminf__le,axiom,
    ! [F: fun_nat_real,G3: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G3,N2)))
     => ( summable_real(F)
       => ( summable_real(G3)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(F)),suminf_real(G3))) ) ) ) ).

% suminf_le
tff(fact_7835_suminf__le,axiom,
    ! [F: fun_nat_nat,G3: fun_nat_nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(G3,N2)))
     => ( summable_nat(F)
       => ( summable_nat(G3)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,suminf_nat(F)),suminf_nat(G3))) ) ) ) ).

% suminf_le
tff(fact_7836_suminf__le,axiom,
    ! [F: fun_nat_int,G3: fun_nat_int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(G3,N2)))
     => ( summable_int(F)
       => ( summable_int(G3)
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,suminf_int(F)),suminf_int(G3))) ) ) ) ).

% suminf_le
tff(fact_7837_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => summable_real(F) ) ) ).

% summable_finite
tff(fact_7838_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => summable_nat(F) ) ) ).

% summable_finite
tff(fact_7839_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => summable_int(F) ) ) ).

% summable_finite
tff(fact_7840_summable__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hd(C),F))
     => ( ( C != zero_zero_complex )
       => summable_complex(F) ) ) ).

% summable_mult_D
tff(fact_7841_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_he(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_7842_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_7843_summable__zero__power,axiom,
    summable_real(aa_real_fun_nat_real(power_power_real,zero_zero_real)) ).

% summable_zero_power
tff(fact_7844_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_7845_powser__insidea,axiom,
    ! [F: fun_nat_real,X: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ho(F),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,X)))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_hp(F),Z)) ) ) ).

% powser_insidea
tff(fact_7846_powser__insidea,axiom,
    ! [F: fun_nat_complex,X: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_hq(F),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,X)))
       => summable_real(aa_com20334259897932645t_real(aTP_Lamm_hr(F),Z)) ) ) ).

% powser_insidea
tff(fact_7847_suminf__nonneg,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),suminf_real(F))) ) ) ).

% suminf_nonneg
tff(fact_7848_suminf__nonneg,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),suminf_nat(F))) ) ) ).

% suminf_nonneg
tff(fact_7849_suminf__nonneg,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),suminf_int(F))) ) ) ).

% suminf_nonneg
tff(fact_7850_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( ( suminf_real(F) = zero_zero_real )
        <=> ! [N3: nat] : aa_nat_real(F,N3) = zero_zero_real ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7851_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( ( suminf_nat(F) = zero_zero_nat )
        <=> ! [N3: nat] : aa_nat_nat(F,N3) = zero_zero_nat ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7852_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( ( suminf_int(F) = zero_zero_int )
        <=> ! [N3: nat] : aa_nat_int(F,N3) = zero_zero_int ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7853_suminf__pos,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,N2)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F))) ) ) ).

% suminf_pos
tff(fact_7854_suminf__pos,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F))) ) ) ).

% suminf_pos
tff(fact_7855_suminf__pos,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,N2)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F))) ) ) ).

% suminf_pos
tff(fact_7856_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_gw(F)) ).

% summable_zero_power'
tff(fact_7857_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_gx(F)) ).

% summable_zero_power'
tff(fact_7858_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_hs(F)) ).

% summable_zero_power'
tff(fact_7859_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_gw(F)) ).

% summable_0_powser
tff(fact_7860_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_gx(F)) ).

% summable_0_powser
tff(fact_7861_summable__norm__comparison__test,axiom,
    ! [F: fun_nat_complex,G3: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G3,N2))) )
     => ( summable_real(G3)
       => summable_real(aTP_Lamm_ht(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_7862_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G3: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N2))),aa_nat_real(G3,N2))) )
     => ( summable_real(G3)
       => summable_real(aTP_Lamm_hu(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_7863_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_hu(F))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(F))),suminf_real(aTP_Lamm_hu(F)))) ) ).

% summable_rabs
tff(fact_7864_tanh__real__lt__1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),one_one_real)) ).

% tanh_real_lt_1
tff(fact_7865_suminf__pos__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F)))
        <=> ? [I4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I4))) ) ) ) ).

% suminf_pos_iff
tff(fact_7866_suminf__pos__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F)))
        <=> ? [I4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I4))) ) ) ) ).

% suminf_pos_iff
tff(fact_7867_suminf__pos__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F)))
        <=> ? [I4: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I4))) ) ) ) ).

% suminf_pos_iff
tff(fact_7868_suminf__pos2,axiom,
    ! [F: fun_nat_real,I: nat] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F))) ) ) ) ).

% suminf_pos2
tff(fact_7869_suminf__pos2,axiom,
    ! [F: fun_nat_nat,I: nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F))) ) ) ) ).

% suminf_pos2
tff(fact_7870_suminf__pos2,axiom,
    ! [F: fun_nat_int,I: nat] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F))) ) ) ) ).

% suminf_pos2
tff(fact_7871_summable__geometric,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
     => summable_real(aa_real_fun_nat_real(power_power_real,C)) ) ).

% summable_geometric
tff(fact_7872_summable__geometric,axiom,
    ! [C: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
     => summable_complex(power_power_complex(C)) ) ).

% summable_geometric
tff(fact_7873_complete__algebra__summable__geometric,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),one_one_real))
     => summable_real(aa_real_fun_nat_real(power_power_real,X)) ) ).

% complete_algebra_summable_geometric
tff(fact_7874_complete__algebra__summable__geometric,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),one_one_real))
     => summable_complex(power_power_complex(X)) ) ).

% complete_algebra_summable_geometric
tff(fact_7875_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_hv(F)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),aa_nat_real(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_7876_round__mono,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,X)),aa_rat_int(archim7778729529865785530nd_rat,Y))) ) ).

% round_mono
tff(fact_7877_summable__norm,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_hw(F))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(F))),suminf_real(aTP_Lamm_hw(F)))) ) ).

% summable_norm
tff(fact_7878_summable__norm,axiom,
    ! [F: fun_nat_complex] :
      ( summable_real(aTP_Lamm_ht(F))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(F))),suminf_real(aTP_Lamm_ht(F)))) ) ).

% summable_norm
tff(fact_7879_ceiling__ge__round,axiom,
    ! [X: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim8280529875227126926d_real,X)),aa_real_int(archim7802044766580827645g_real,X))) ).

% ceiling_ge_round
tff(fact_7880_tanh__real__gt__neg1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(tanh_real,X))) ).

% tanh_real_gt_neg1
tff(fact_7881_powser__inside,axiom,
    ! [F: fun_nat_real,X: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ho(F),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,X)))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_ho(F),Z)) ) ) ).

% powser_inside
tff(fact_7882_powser__inside,axiom,
    ! [F: fun_nat_complex,X: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_hq(F),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,X)))
       => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_hq(F),Z)) ) ) ).

% powser_inside
tff(fact_7883_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_hq(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_hq(F),Z)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(F,zero_zero_nat)),aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_hx(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_7884_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ho(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_ho(F),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,zero_zero_nat)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_hy(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_7885_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_hq(F),Z))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_hx(F),Z))),Z) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_hq(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7886_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ho(F),Z))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_hy(F),Z))),Z) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_ho(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7887_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
     => ( summable_real(F)
       => ? [N8: nat] :
          ! [N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_hz(F),N5)))),R2)) ) ) ) ).

% suminf_exist_split
tff(fact_7888_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
     => ( summable_complex(F)
       => ? [N8: nat] :
          ! [N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ia(F),N5)))),R2)) ) ) ) ).

% suminf_exist_split
tff(fact_7889_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I2)),one_one_real))
     => ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Z))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),one_one_real))
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_ho(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_7890_Abel__lemma,axiom,
    ! [R2: real,R0: real,A2: fun_nat_complex,M5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R2),R0))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(A2,N2))),aa_nat_real(aa_real_fun_nat_real(power_power_real,R0),N2))),M5))
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_ib(R2),A2)) ) ) ) ).

% Abel_lemma
tff(fact_7891_summable__ratio__test,axiom,
    ! [C: real,N4: nat,F: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),one_one_real))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N4),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,aa_nat_nat(suc,N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))))) )
       => summable_real(F) ) ) ).

% summable_ratio_test
tff(fact_7892_summable__ratio__test,axiom,
    ! [C: real,N4: nat,F: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),one_one_real))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N4),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,aa_nat_nat(suc,N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))))) )
       => summable_complex(F) ) ) ).

% summable_ratio_test
tff(fact_7893_round__diff__minimal,axiom,
    ! [Z: real,M: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Z))))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),aa_int_real(ring_1_of_int_real,M))))) ).

% round_diff_minimal
tff(fact_7894_round__diff__minimal,axiom,
    ! [Z: rat,M: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Z),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Z))))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Z),aa_int_rat(ring_1_of_int_rat,M))))) ).

% round_diff_minimal
tff(fact_7895_of__int__round__le,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% of_int_round_le
tff(fact_7896_of__int__round__le,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))) ).

% of_int_round_le
tff(fact_7897_of__int__round__ge,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X)))) ).

% of_int_round_ge
tff(fact_7898_of__int__round__ge,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X)))) ).

% of_int_round_ge
tff(fact_7899_of__int__round__gt,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X)))) ).

% of_int_round_gt
tff(fact_7900_of__int__round__gt,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X)))) ).

% of_int_round_gt
tff(fact_7901_of__int__round__abs__le,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% of_int_round_abs_le
tff(fact_7902_of__int__round__abs__le,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))),X))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_abs_le
tff(fact_7903_accp__subset,axiom,
    ! [R1: fun_Pr1775001904083774893t_bool,R22: fun_Pr1775001904083774893t_bool] :
      ( ord_le2118279620499019085t_bool(R1,R22)
     => ord_le8032738712617130713t_bool(accp_P8646395344606611882on_nat(R22),accp_P8646395344606611882on_nat(R1)) ) ).

% accp_subset
tff(fact_7904_accp__subset,axiom,
    ! [R1: fun_Pr248683899324676269t_bool,R22: fun_Pr248683899324676269t_bool] :
      ( ord_le2087689725136068685t_bool(R1,R22)
     => ord_le2468614981408567801t_bool(accp_P4275260045618599050at_nat(R22),accp_P4275260045618599050at_nat(R1)) ) ).

% accp_subset
tff(fact_7905_accp__subset,axiom,
    ! [R1: fun_Pr3099820731930151597t_bool,R22: fun_Pr3099820731930151597t_bool] :
      ( ord_le4938826557741544013t_bool(R1,R22)
     => ord_le4737936599360737601t_bool(accp_P1096762738010456898nt_int(R22),accp_P1096762738010456898nt_int(R1)) ) ).

% accp_subset
tff(fact_7906_accp__subset,axiom,
    ! [R1: fun_li8598321315282354825t_bool,R22: fun_li8598321315282354825t_bool] :
      ( ord_le4434192926011882025t_bool(R1,R22)
     => ord_le2936377143570524336t_bool(accp_list_nat(R22),accp_list_nat(R1)) ) ).

% accp_subset
tff(fact_7907_accp__subset,axiom,
    ! [R1: fun_nat_fun_nat_bool,R22: fun_nat_fun_nat_bool] :
      ( ord_le4899916226251588169t_bool(R1,R22)
     => ord_le8840051076831633728t_bool(accp_nat(R22),accp_nat(R1)) ) ).

% accp_subset
tff(fact_7908_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_bool,X: fun_bool_complex,Y: fun_bool_complex] :
      ( pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_fun4000492451996535199l_bool(aTP_Lamm_ic(I5),X))))
     => ( pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_fun4000492451996535199l_bool(aTP_Lamm_ic(I5),Y))))
       => pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_fun4000492451996535199l_bool(aa_fun492156817397607122l_bool(aTP_Lamm_id(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_7909_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_complex,Y: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_ie(I5),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_ie(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_if(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_7910_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_complex,Y: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_ig(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_ig(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_ih(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_7911_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_complex,Y: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_ii(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_ii(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_ij(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_7912_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_complex,Y: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ik(I5),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ik(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_il(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_7913_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X: fun_Ex3741802288643809319omplex,Y: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun7434105802671982559t_bool(aTP_Lamm_im(I5),X))))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun7434105802671982559t_bool(aTP_Lamm_im(I5),Y))))
       => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun7434105802671982559t_bool(aa_fun6445447259835133315t_bool(aTP_Lamm_in(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_7914_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_bool,X: fun_bool_real,Y: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_fun2512541629346179229l_bool(aTP_Lamm_io(I5),X))))
     => ( pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_fun2512541629346179229l_bool(aTP_Lamm_io(I5),Y))))
       => pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_fun2512541629346179229l_bool(aa_fun8125513006060278610l_bool(aTP_Lamm_ip(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_7915_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_real,Y: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_iq(I5),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_iq(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_ir(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_7916_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_real,Y: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_is(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_is(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_it(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_7917_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_real,Y: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_iu(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_iu(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_iv(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_7918_sum__gp,axiom,
    ! [N: nat,M: nat,X: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2906978787729119204at_rat(power_power_rat(X),set_or1269000886237332187st_nat(M,N)) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X = one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X),set_or1269000886237332187st_nat(M,N)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) )
          & ( ( X != one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X),set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),M)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N)))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_7919_sum__gp,axiom,
    ! [N: nat,M: nat,X: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),set_or1269000886237332187st_nat(M,N)) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X = one_one_complex )
           => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),set_or1269000886237332187st_nat(M,N)) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) )
          & ( ( X != one_one_complex )
           => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),set_or1269000886237332187st_nat(M,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N)))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_7920_sum__gp,axiom,
    ! [N: nat,M: nat,X: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),set_or1269000886237332187st_nat(M,N)) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X = one_one_real )
           => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),set_or1269000886237332187st_nat(M,N)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) )
          & ( ( X != one_one_real )
           => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),set_or1269000886237332187st_nat(M,N)) = 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,X),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_7921_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_bool,X: fun_bool_real,Y: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_fun2512541629346179229l_bool(aTP_Lamm_iw(I5),X))))
     => ( pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_fun2512541629346179229l_bool(aTP_Lamm_iw(I5),Y))))
       => pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_fun2512541629346179229l_bool(aa_fun8125513006060278610l_bool(aTP_Lamm_ix(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7922_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_real,Y: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_iy(I5),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_iy(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_iz(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7923_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_real,Y: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_ja(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_ja(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_jb(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7924_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_real,Y: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_jc(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_jc(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_jd(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7925_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_real,Y: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_je(I5),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_je(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_jf(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7926_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X: fun_Ex1451620480498224165t_real,Y: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun338017137003533149t_bool(aTP_Lamm_jg(I5),X))))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun338017137003533149t_bool(aTP_Lamm_jg(I5),Y))))
       => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun338017137003533149t_bool(aa_fun6293361260138588547t_bool(aTP_Lamm_jh(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7927_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_bool,X: fun_bool_rat,Y: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_fun4510340986217973961l_bool(aTP_Lamm_ji(I5),X))))
     => ( pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_fun4510340986217973961l_bool(aTP_Lamm_ji(I5),Y))))
       => pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_fun4510340986217973961l_bool(aa_fun5608642267463484754l_bool(aTP_Lamm_jj(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7928_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_rat,Y: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_jk(I5),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_jk(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_jl(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7929_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_rat,Y: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_jm(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_jm(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_jn(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7930_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_rat,Y: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun258058769140594373t_bool(aTP_Lamm_jo(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun258058769140594373t_bool(aTP_Lamm_jo(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun258058769140594373t_bool(aa_fun465274253410646749t_bool(aTP_Lamm_jp(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7931_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_jq,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% geometric_deriv_sums
tff(fact_7932_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real))
     => pp(aa_complex_bool(sums_complex(aTP_Lamm_jr(Z)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% geometric_deriv_sums
tff(fact_7933_log__base__10__eq1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,X)) ) ) ).

% log_base_10_eq1
tff(fact_7934_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(80)
tff(fact_7935_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(73)
tff(fact_7936_sum_Oneutral__const,axiom,
    ! [A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cc),A3) = zero_zero_nat ).

% sum.neutral_const
tff(fact_7937_sum_Oneutral__const,axiom,
    ! [A3: set_complex] : groups7754918857620584856omplex(aTP_Lamm_bz,A3) = zero_zero_complex ).

% sum.neutral_const
tff(fact_7938_sum_Oneutral__const,axiom,
    ! [A3: set_int] : groups4538972089207619220nt_int(aTP_Lamm_cd,A3) = zero_zero_int ).

% sum.neutral_const
tff(fact_7939_sum_Oneutral__const,axiom,
    ! [A3: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gy),A3) = zero_zero_real ).

% sum.neutral_const
tff(fact_7940_sums__zero,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_gy),zero_zero_real)) ).

% sums_zero
tff(fact_7941_sums__zero,axiom,
    pp(aa_nat_bool(sums_nat(aTP_Lamm_cc),zero_zero_nat)) ).

% sums_zero
tff(fact_7942_sums__zero,axiom,
    pp(aa_int_bool(sums_int(aTP_Lamm_gz),zero_zero_int)) ).

% sums_zero
tff(fact_7943_sum_Oempty,axiom,
    ! [G3: fun_real_real] : groups8097168146408367636l_real(G3,bot_bot_set_real) = zero_zero_real ).

% sum.empty
tff(fact_7944_sum_Oempty,axiom,
    ! [G3: fun_real_rat] : groups1300246762558778688al_rat(G3,bot_bot_set_real) = zero_zero_rat ).

% sum.empty
tff(fact_7945_sum_Oempty,axiom,
    ! [G3: fun_real_nat] : groups1935376822645274424al_nat(G3,bot_bot_set_real) = zero_zero_nat ).

% sum.empty
tff(fact_7946_sum_Oempty,axiom,
    ! [G3: fun_real_int] : groups1932886352136224148al_int(G3,bot_bot_set_real) = zero_zero_int ).

% sum.empty
tff(fact_7947_sum_Oempty,axiom,
    ! [G3: fun_bool_real] : groups502887708668150281l_real(G3,bot_bot_set_bool) = zero_zero_real ).

% sum.empty
tff(fact_7948_sum_Oempty,axiom,
    ! [G3: fun_bool_rat] : groups5133371213642249141ol_rat(G3,bot_bot_set_bool) = zero_zero_rat ).

% sum.empty
tff(fact_7949_sum_Oempty,axiom,
    ! [G3: fun_bool_nat] : groups5768501273728744877ol_nat(G3,bot_bot_set_bool) = zero_zero_nat ).

% sum.empty
tff(fact_7950_sum_Oempty,axiom,
    ! [G3: fun_bool_int] : groups5766010803219694601ol_int(G3,bot_bot_set_bool) = zero_zero_int ).

% sum.empty
tff(fact_7951_sum_Oempty,axiom,
    ! [G3: fun_nat_rat] : groups2906978787729119204at_rat(G3,bot_bot_set_nat) = zero_zero_rat ).

% sum.empty
tff(fact_7952_sum_Oempty,axiom,
    ! [G3: fun_nat_int] : groups3539618377306564664at_int(G3,bot_bot_set_nat) = zero_zero_int ).

% sum.empty
tff(fact_7953_sum_Oinfinite,axiom,
    ! [A3: set_int,G3: fun_int_real] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups8778361861064173332t_real(G3,A3) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_7954_sum_Oinfinite,axiom,
    ! [A3: set_complex,G3: fun_complex_real] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5808333547571424918x_real(G3,A3) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_7955_sum_Oinfinite,axiom,
    ! [A3: set_Extended_enat,G3: fun_Ex1451620480498224165t_real] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups4148127829035722712t_real(G3,A3) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_7956_sum_Oinfinite,axiom,
    ! [A3: set_nat,G3: fun_nat_rat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( groups2906978787729119204at_rat(G3,A3) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_7957_sum_Oinfinite,axiom,
    ! [A3: set_int,G3: fun_int_rat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups3906332499630173760nt_rat(G3,A3) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_7958_sum_Oinfinite,axiom,
    ! [A3: set_complex,G3: fun_complex_rat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5058264527183730370ex_rat(G3,A3) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_7959_sum_Oinfinite,axiom,
    ! [A3: set_Extended_enat,G3: fun_Ex4152069839811072081at_rat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups1392844769737527556at_rat(G3,A3) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_7960_sum_Oinfinite,axiom,
    ! [A3: set_int,G3: fun_int_nat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups4541462559716669496nt_nat(G3,A3) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_7961_sum_Oinfinite,axiom,
    ! [A3: set_complex,G3: fun_complex_nat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5693394587270226106ex_nat(G3,A3) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_7962_sum_Oinfinite,axiom,
    ! [A3: set_Extended_enat,G3: fun_Ex7660298398204468809at_nat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups2027974829824023292at_nat(G3,A3) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_7963_sum__eq__0__iff,axiom,
    ! [F2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,F2))
     => ( ( groups4541462559716669496nt_nat(F,F2) = zero_zero_nat )
      <=> ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),F2))
           => ( aa_int_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_7964_sum__eq__0__iff,axiom,
    ! [F2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F2))
     => ( ( groups5693394587270226106ex_nat(F,F2) = zero_zero_nat )
      <=> ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),F2))
           => ( aa_complex_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_7965_sum__eq__0__iff,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F2))
     => ( ( aa_set4749074711808745205at_nat(groups977919841031483927at_nat(F),F2) = zero_zero_nat )
      <=> ! [X3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),F2))
           => ( aa_Pro4243173770044397973at_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_7966_sum__eq__0__iff,axiom,
    ! [F2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F2))
     => ( ( groups2027974829824023292at_nat(F,F2) = zero_zero_nat )
      <=> ! [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),F2))
           => ( aa_Extended_enat_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_7967_sum__eq__0__iff,axiom,
    ! [F2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F2))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),F2) = zero_zero_nat )
      <=> ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),F2))
           => ( aa_nat_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_7968_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(81)
tff(fact_7969_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(72)
tff(fact_7970_semiring__norm_I77_J,axiom,
    ! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),aa_num_num(bit1,N))) ).

% semiring_norm(77)
tff(fact_7971_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M)),one)) ).

% semiring_norm(70)
tff(fact_7972_sum_Odelta_H,axiom,
    ! [S: set_real,A2: real,B3: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_js(A2),B3),S) = aa_real_real(B3,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_js(A2),B3),S) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_7973_sum_Odelta_H,axiom,
    ! [S: set_bool,A2: bool,B3: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( ( pp(aa_set_bool_bool(member_bool(A2),S))
         => ( groups502887708668150281l_real(aa_fun8933258245593490108l_real(aTP_Lamm_jt(A2),B3),S) = aa_bool_real(B3,A2) ) )
        & ( ~ pp(aa_set_bool_bool(member_bool(A2),S))
         => ( groups502887708668150281l_real(aa_fun8933258245593490108l_real(aTP_Lamm_jt(A2),B3),S) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_7974_sum_Odelta_H,axiom,
    ! [S: set_int,A2: int,B3: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ( pp(aa_set_int_bool(member_int(A2),S))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ju(A2),B3),S) = aa_int_real(B3,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A2),S))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ju(A2),B3),S) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_7975_sum_Odelta_H,axiom,
    ! [S: set_complex,A2: complex,B3: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_jv(A2),B3),S) = aa_complex_real(B3,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_jv(A2),B3),S) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_7976_sum_Odelta_H,axiom,
    ! [S: set_Extended_enat,A2: extended_enat,B3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),S))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_jw(A2),B3),S) = aa_Ext2541572767299617110t_real(B3,A2) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),S))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_jw(A2),B3),S) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_7977_sum_Odelta_H,axiom,
    ! [S: set_real,A2: real,B3: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_jx(A2),B3),S) = aa_real_rat(B3,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_jx(A2),B3),S) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_7978_sum_Odelta_H,axiom,
    ! [S: set_bool,A2: bool,B3: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( ( pp(aa_set_bool_bool(member_bool(A2),S))
         => ( groups5133371213642249141ol_rat(aa_fun224000846028461884ol_rat(aTP_Lamm_jy(A2),B3),S) = aa_bool_rat(B3,A2) ) )
        & ( ~ pp(aa_set_bool_bool(member_bool(A2),S))
         => ( groups5133371213642249141ol_rat(aa_fun224000846028461884ol_rat(aTP_Lamm_jy(A2),B3),S) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_7979_sum_Odelta_H,axiom,
    ! [S: set_nat,A2: nat,B3: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ( pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_jz(A2),B3),S) = aa_nat_rat(B3,A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_jz(A2),B3),S) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_7980_sum_Odelta_H,axiom,
    ! [S: set_int,A2: int,B3: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ( pp(aa_set_int_bool(member_int(A2),S))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ka(A2),B3),S) = aa_int_rat(B3,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A2),S))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ka(A2),B3),S) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_7981_sum_Odelta_H,axiom,
    ! [S: set_complex,A2: complex,B3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_kb(A2),B3),S) = aa_complex_rat(B3,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_kb(A2),B3),S) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_7982_sum_Odelta,axiom,
    ! [S: set_real,A2: real,B3: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_kc(A2),B3),S) = aa_real_real(B3,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_kc(A2),B3),S) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_7983_sum_Odelta,axiom,
    ! [S: set_bool,A2: bool,B3: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( ( pp(aa_set_bool_bool(member_bool(A2),S))
         => ( groups502887708668150281l_real(aa_fun8933258245593490108l_real(aTP_Lamm_kd(A2),B3),S) = aa_bool_real(B3,A2) ) )
        & ( ~ pp(aa_set_bool_bool(member_bool(A2),S))
         => ( groups502887708668150281l_real(aa_fun8933258245593490108l_real(aTP_Lamm_kd(A2),B3),S) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_7984_sum_Odelta,axiom,
    ! [S: set_int,A2: int,B3: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ( pp(aa_set_int_bool(member_int(A2),S))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ke(A2),B3),S) = aa_int_real(B3,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A2),S))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ke(A2),B3),S) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_7985_sum_Odelta,axiom,
    ! [S: set_complex,A2: complex,B3: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_kf(A2),B3),S) = aa_complex_real(B3,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_kf(A2),B3),S) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_7986_sum_Odelta,axiom,
    ! [S: set_Extended_enat,A2: extended_enat,B3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),S))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_kg(A2),B3),S) = aa_Ext2541572767299617110t_real(B3,A2) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),S))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_kg(A2),B3),S) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_7987_sum_Odelta,axiom,
    ! [S: set_real,A2: real,B3: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_kh(A2),B3),S) = aa_real_rat(B3,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_kh(A2),B3),S) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_7988_sum_Odelta,axiom,
    ! [S: set_bool,A2: bool,B3: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( ( pp(aa_set_bool_bool(member_bool(A2),S))
         => ( groups5133371213642249141ol_rat(aa_fun224000846028461884ol_rat(aTP_Lamm_ki(A2),B3),S) = aa_bool_rat(B3,A2) ) )
        & ( ~ pp(aa_set_bool_bool(member_bool(A2),S))
         => ( groups5133371213642249141ol_rat(aa_fun224000846028461884ol_rat(aTP_Lamm_ki(A2),B3),S) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_7989_sum_Odelta,axiom,
    ! [S: set_nat,A2: nat,B3: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ( pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_kj(A2),B3),S) = aa_nat_rat(B3,A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_kj(A2),B3),S) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_7990_sum_Odelta,axiom,
    ! [S: set_int,A2: int,B3: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ( pp(aa_set_int_bool(member_int(A2),S))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_kk(A2),B3),S) = aa_int_rat(B3,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A2),S))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_kk(A2),B3),S) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_7991_sum_Odelta,axiom,
    ! [S: set_complex,A2: complex,B3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_kl(A2),B3),S) = aa_complex_rat(B3,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_kl(A2),B3),S) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_7992_sum__abs,axiom,
    ! [F: fun_int_int,A3: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A3))),groups4538972089207619220nt_int(aTP_Lamm_km(F),A3))) ).

% sum_abs
tff(fact_7993_sum__abs,axiom,
    ! [F: fun_nat_real,A3: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_set_nat_real(groups6591440286371151544t_real(F),A3))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hu(F)),A3))) ).

% sum_abs
tff(fact_7994_sum_Oinsert,axiom,
    ! [A3: set_real,X: real,G3: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ~ pp(aa_set_real_bool(member_real(X),A3))
       => ( groups8097168146408367636l_real(G3,insert_real(X,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G3,X)),groups8097168146408367636l_real(G3,A3)) ) ) ) ).

% sum.insert
tff(fact_7995_sum_Oinsert,axiom,
    ! [A3: set_bool,X: bool,G3: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( ~ pp(aa_set_bool_bool(member_bool(X),A3))
       => ( groups502887708668150281l_real(G3,insert_bool(X,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_bool_real(G3,X)),groups502887708668150281l_real(G3,A3)) ) ) ) ).

% sum.insert
tff(fact_7996_sum_Oinsert,axiom,
    ! [A3: set_int,X: int,G3: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ~ pp(aa_set_int_bool(member_int(X),A3))
       => ( groups8778361861064173332t_real(G3,insert_int(X,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G3,X)),groups8778361861064173332t_real(G3,A3)) ) ) ) ).

% sum.insert
tff(fact_7997_sum_Oinsert,axiom,
    ! [A3: set_complex,X: complex,G3: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A3))
       => ( groups5808333547571424918x_real(G3,insert_complex(X,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G3,X)),groups5808333547571424918x_real(G3,A3)) ) ) ) ).

% sum.insert
tff(fact_7998_sum_Oinsert,axiom,
    ! [A3: set_Extended_enat,X: extended_enat,G3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A3))
       => ( groups4148127829035722712t_real(G3,insert_Extended_enat(X,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Ext2541572767299617110t_real(G3,X)),groups4148127829035722712t_real(G3,A3)) ) ) ) ).

% sum.insert
tff(fact_7999_sum_Oinsert,axiom,
    ! [A3: set_real,X: real,G3: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ~ pp(aa_set_real_bool(member_real(X),A3))
       => ( groups1300246762558778688al_rat(G3,insert_real(X,A3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(G3,X)),groups1300246762558778688al_rat(G3,A3)) ) ) ) ).

% sum.insert
tff(fact_8000_sum_Oinsert,axiom,
    ! [A3: set_bool,X: bool,G3: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( ~ pp(aa_set_bool_bool(member_bool(X),A3))
       => ( groups5133371213642249141ol_rat(G3,insert_bool(X,A3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_bool_rat(G3,X)),groups5133371213642249141ol_rat(G3,A3)) ) ) ) ).

% sum.insert
tff(fact_8001_sum_Oinsert,axiom,
    ! [A3: set_nat,X: nat,G3: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),A3))
       => ( groups2906978787729119204at_rat(G3,insert_nat(X,A3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G3,X)),groups2906978787729119204at_rat(G3,A3)) ) ) ) ).

% sum.insert
tff(fact_8002_sum_Oinsert,axiom,
    ! [A3: set_int,X: int,G3: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ~ pp(aa_set_int_bool(member_int(X),A3))
       => ( groups3906332499630173760nt_rat(G3,insert_int(X,A3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(G3,X)),groups3906332499630173760nt_rat(G3,A3)) ) ) ) ).

% sum.insert
tff(fact_8003_sum_Oinsert,axiom,
    ! [A3: set_complex,X: complex,G3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A3))
       => ( groups5058264527183730370ex_rat(G3,insert_complex(X,A3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(G3,X)),groups5058264527183730370ex_rat(G3,A3)) ) ) ) ).

% sum.insert
tff(fact_8004_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A3: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_km(F),A3))) ).

% sum_abs_ge_zero
tff(fact_8005_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A3: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hu(F)),A3))) ).

% sum_abs_ge_zero
tff(fact_8006_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(74)
tff(fact_8007_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(79)
tff(fact_8008_powser__sums__zero__iff,axiom,
    ! [A2: fun_nat_complex,X: complex] :
      ( pp(aa_complex_bool(sums_complex(aTP_Lamm_gw(A2)),X))
    <=> ( aa_nat_complex(A2,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_8009_powser__sums__zero__iff,axiom,
    ! [A2: fun_nat_real,X: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_gx(A2)),X))
    <=> ( aa_nat_real(A2,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_8010_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G3: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups2906978787729119204at_rat(G3,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups2906978787729119204at_rat(G3,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G3,set_or1269000886237332187st_nat(M,N))),aa_nat_rat(G3,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_8011_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G3: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups3539618377306564664at_int(G3,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = zero_zero_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups3539618377306564664at_int(G3,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G3,set_or1269000886237332187st_nat(M,N))),aa_nat_int(G3,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_8012_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G3: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G3),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G3),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G3),set_or1269000886237332187st_nat(M,N))),aa_nat_nat(G3,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_8013_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G3: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_real(groups6591440286371151544t_real(G3),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_real(groups6591440286371151544t_real(G3),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G3),set_or1269000886237332187st_nat(M,N))),aa_nat_real(G3,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_8014_sum__zero__power,axiom,
    ! [A3: set_nat,C: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A3))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_gw(C)),A3) = aa_nat_complex(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A3))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_gw(C)),A3) = zero_zero_complex ) ) ) ).

% sum_zero_power
tff(fact_8015_sum__zero__power,axiom,
    ! [A3: set_nat,C: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A3))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_kn(C),A3) = aa_nat_rat(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A3))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_kn(C),A3) = zero_zero_rat ) ) ) ).

% sum_zero_power
tff(fact_8016_sum__zero__power,axiom,
    ! [A3: set_nat,C: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A3))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gx(C)),A3) = aa_nat_real(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A3))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gx(C)),A3) = zero_zero_real ) ) ) ).

% sum_zero_power
tff(fact_8017_sum__zero__power_H,axiom,
    ! [A3: set_nat,C: fun_nat_rat,D: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A3))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ko(C),D),A3) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(C,zero_zero_nat)),aa_nat_rat(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A3))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ko(C),D),A3) = zero_zero_rat ) ) ) ).

% sum_zero_power'
tff(fact_8018_sum__zero__power_H,axiom,
    ! [A3: set_nat,C: fun_nat_complex,D: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A3))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_kp(C),D)),A3) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(C,zero_zero_nat)),aa_nat_complex(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A3))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_kp(C),D)),A3) = zero_zero_complex ) ) ) ).

% sum_zero_power'
tff(fact_8019_sum__zero__power_H,axiom,
    ! [A3: set_nat,C: fun_nat_real,D: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A3))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_kq(C),D)),A3) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(C,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A3))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_kq(C),D)),A3) = zero_zero_real ) ) ) ).

% sum_zero_power'
tff(fact_8020_sums__le,axiom,
    ! [F: fun_nat_nat,G3: fun_nat_nat,S2: nat,T: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(G3,N2)))
     => ( pp(aa_nat_bool(sums_nat(F),S2))
       => ( pp(aa_nat_bool(sums_nat(G3),T))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,S2),T)) ) ) ) ).

% sums_le
tff(fact_8021_sums__le,axiom,
    ! [F: fun_nat_int,G3: fun_nat_int,S2: int,T: int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(G3,N2)))
     => ( pp(aa_int_bool(sums_int(F),S2))
       => ( pp(aa_int_bool(sums_int(G3),T))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,S2),T)) ) ) ) ).

% sums_le
tff(fact_8022_sums__0,axiom,
    ! [F: fun_nat_real] :
      ( ! [N2: nat] : aa_nat_real(F,N2) = zero_zero_real
     => pp(aa_real_bool(sums_real(F),zero_zero_real)) ) ).

% sums_0
tff(fact_8023_sums__0,axiom,
    ! [F: fun_nat_nat] :
      ( ! [N2: nat] : aa_nat_nat(F,N2) = zero_zero_nat
     => pp(aa_nat_bool(sums_nat(F),zero_zero_nat)) ) ).

% sums_0
tff(fact_8024_sums__0,axiom,
    ! [F: fun_nat_int] :
      ( ! [N2: nat] : aa_nat_int(F,N2) = zero_zero_int
     => pp(aa_int_bool(sums_int(F),zero_zero_int)) ) ).

% sums_0
tff(fact_8025_sum_Oneutral,axiom,
    ! [A3: set_nat,G3: fun_nat_nat] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),A3))
         => ( aa_nat_nat(G3,X5) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G3),A3) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_8026_sum_Oneutral,axiom,
    ! [A3: set_complex,G3: fun_complex_complex] :
      ( ! [X5: complex] :
          ( pp(aa_set_complex_bool(member_complex(X5),A3))
         => ( aa_complex_complex(G3,X5) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G3,A3) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_8027_sum_Oneutral,axiom,
    ! [A3: set_int,G3: fun_int_int] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),A3))
         => ( aa_int_int(G3,X5) = zero_zero_int ) )
     => ( groups4538972089207619220nt_int(G3,A3) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_8028_sum_Oneutral,axiom,
    ! [A3: set_nat,G3: fun_nat_real] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),A3))
         => ( aa_nat_real(G3,X5) = zero_zero_real ) )
     => ( aa_set_nat_real(groups6591440286371151544t_real(G3),A3) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_8029_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: fun_real_real,A3: set_real] :
      ( ( groups8097168146408367636l_real(G3,A3) != zero_zero_real )
     => ~ ! [A: real] :
            ( pp(aa_set_real_bool(member_real(A),A3))
           => ( aa_real_real(G3,A) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8030_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: fun_bool_real,A3: set_bool] :
      ( ( groups502887708668150281l_real(G3,A3) != zero_zero_real )
     => ~ ! [A: bool] :
            ( pp(aa_set_bool_bool(member_bool(A),A3))
           => ( aa_bool_real(G3,A) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8031_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: fun_int_real,A3: set_int] :
      ( ( groups8778361861064173332t_real(G3,A3) != zero_zero_real )
     => ~ ! [A: int] :
            ( pp(aa_set_int_bool(member_int(A),A3))
           => ( aa_int_real(G3,A) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8032_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: fun_real_rat,A3: set_real] :
      ( ( groups1300246762558778688al_rat(G3,A3) != zero_zero_rat )
     => ~ ! [A: real] :
            ( pp(aa_set_real_bool(member_real(A),A3))
           => ( aa_real_rat(G3,A) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8033_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: fun_bool_rat,A3: set_bool] :
      ( ( groups5133371213642249141ol_rat(G3,A3) != zero_zero_rat )
     => ~ ! [A: bool] :
            ( pp(aa_set_bool_bool(member_bool(A),A3))
           => ( aa_bool_rat(G3,A) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8034_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: fun_nat_rat,A3: set_nat] :
      ( ( groups2906978787729119204at_rat(G3,A3) != zero_zero_rat )
     => ~ ! [A: nat] :
            ( pp(aa_set_nat_bool(member_nat(A),A3))
           => ( aa_nat_rat(G3,A) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8035_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: fun_int_rat,A3: set_int] :
      ( ( groups3906332499630173760nt_rat(G3,A3) != zero_zero_rat )
     => ~ ! [A: int] :
            ( pp(aa_set_int_bool(member_int(A),A3))
           => ( aa_int_rat(G3,A) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8036_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: fun_real_nat,A3: set_real] :
      ( ( groups1935376822645274424al_nat(G3,A3) != zero_zero_nat )
     => ~ ! [A: real] :
            ( pp(aa_set_real_bool(member_real(A),A3))
           => ( aa_real_nat(G3,A) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8037_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: fun_bool_nat,A3: set_bool] :
      ( ( groups5768501273728744877ol_nat(G3,A3) != zero_zero_nat )
     => ~ ! [A: bool] :
            ( pp(aa_set_bool_bool(member_bool(A),A3))
           => ( aa_bool_nat(G3,A) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8038_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: fun_int_nat,A3: set_int] :
      ( ( groups4541462559716669496nt_nat(G3,A3) != zero_zero_nat )
     => ~ ! [A: int] :
            ( pp(aa_set_int_bool(member_int(A),A3))
           => ( aa_int_nat(G3,A) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8039_sum__norm__le,axiom,
    ! [S: set_real,F: fun_real_complex,G3: fun_real_real] :
      ( ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(F,X5))),aa_real_real(G3,X5))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups5754745047067104278omplex(F,S))),groups8097168146408367636l_real(G3,S))) ) ).

% sum_norm_le
tff(fact_8040_sum__norm__le,axiom,
    ! [S: set_bool,F: fun_bool_complex,G3: fun_bool_real] :
      ( ! [X5: bool] :
          ( pp(aa_set_bool_bool(member_bool(X5),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_bool_complex(F,X5))),aa_bool_real(G3,X5))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups2959343415743557771omplex(F,S))),groups502887708668150281l_real(G3,S))) ) ).

% sum_norm_le
tff(fact_8041_sum__norm__le,axiom,
    ! [S: set_set_nat,F: fun_set_nat_complex,G3: fun_set_nat_real] :
      ( ! [X5: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X5),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(F,X5))),aa_set_nat_real(G3,X5))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups8255218700646806128omplex(F,S))),groups5107569545109728110t_real(G3,S))) ) ).

% sum_norm_le
tff(fact_8042_sum__norm__le,axiom,
    ! [S: set_int,F: fun_int_complex,G3: fun_int_real] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(F,X5))),aa_int_real(G3,X5))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups3049146728041665814omplex(F,S))),groups8778361861064173332t_real(G3,S))) ) ).

% sum_norm_le
tff(fact_8043_sum__norm__le,axiom,
    ! [S: set_nat,F: fun_nat_complex,G3: fun_nat_real] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,X5))),aa_nat_real(G3,X5))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),S))),aa_set_nat_real(groups6591440286371151544t_real(G3),S))) ) ).

% sum_norm_le
tff(fact_8044_sum__norm__le,axiom,
    ! [S: set_complex,F: fun_complex_complex,G3: fun_complex_real] :
      ( ! [X5: complex] :
          ( pp(aa_set_complex_bool(member_complex(X5),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(F,X5))),aa_complex_real(G3,X5))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,S))),groups5808333547571424918x_real(G3,S))) ) ).

% sum_norm_le
tff(fact_8045_sum__norm__le,axiom,
    ! [S: set_nat,F: fun_nat_real,G3: fun_nat_real] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,X5))),aa_nat_real(G3,X5))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),S))),aa_set_nat_real(groups6591440286371151544t_real(G3),S))) ) ).

% sum_norm_le
tff(fact_8046_sum__mono,axiom,
    ! [K4: set_real,F: fun_real_rat,G3: fun_real_rat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),K4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G3,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,K4)),groups1300246762558778688al_rat(G3,K4))) ) ).

% sum_mono
tff(fact_8047_sum__mono,axiom,
    ! [K4: set_bool,F: fun_bool_rat,G3: fun_bool_rat] :
      ( ! [I2: bool] :
          ( pp(aa_set_bool_bool(member_bool(I2),K4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_bool_rat(F,I2)),aa_bool_rat(G3,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5133371213642249141ol_rat(F,K4)),groups5133371213642249141ol_rat(G3,K4))) ) ).

% sum_mono
tff(fact_8048_sum__mono,axiom,
    ! [K4: set_nat,F: fun_nat_rat,G3: fun_nat_rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),K4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G3,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,K4)),groups2906978787729119204at_rat(G3,K4))) ) ).

% sum_mono
tff(fact_8049_sum__mono,axiom,
    ! [K4: set_int,F: fun_int_rat,G3: fun_int_rat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),K4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G3,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K4)),groups3906332499630173760nt_rat(G3,K4))) ) ).

% sum_mono
tff(fact_8050_sum__mono,axiom,
    ! [K4: set_real,F: fun_real_nat,G3: fun_real_nat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),K4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G3,I2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,K4)),groups1935376822645274424al_nat(G3,K4))) ) ).

% sum_mono
tff(fact_8051_sum__mono,axiom,
    ! [K4: set_bool,F: fun_bool_nat,G3: fun_bool_nat] :
      ( ! [I2: bool] :
          ( pp(aa_set_bool_bool(member_bool(I2),K4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(F,I2)),aa_bool_nat(G3,I2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups5768501273728744877ol_nat(F,K4)),groups5768501273728744877ol_nat(G3,K4))) ) ).

% sum_mono
tff(fact_8052_sum__mono,axiom,
    ! [K4: set_int,F: fun_int_nat,G3: fun_int_nat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),K4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G3,I2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K4)),groups4541462559716669496nt_nat(G3,K4))) ) ).

% sum_mono
tff(fact_8053_sum__mono,axiom,
    ! [K4: set_real,F: fun_real_int,G3: fun_real_int] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),K4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G3,I2))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1932886352136224148al_int(F,K4)),groups1932886352136224148al_int(G3,K4))) ) ).

% sum_mono
tff(fact_8054_sum__mono,axiom,
    ! [K4: set_bool,F: fun_bool_int,G3: fun_bool_int] :
      ( ! [I2: bool] :
          ( pp(aa_set_bool_bool(member_bool(I2),K4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(F,I2)),aa_bool_int(G3,I2))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups5766010803219694601ol_int(F,K4)),groups5766010803219694601ol_int(G3,K4))) ) ).

% sum_mono
tff(fact_8055_sum__mono,axiom,
    ! [K4: set_nat,F: fun_nat_int,G3: fun_nat_int] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),K4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G3,I2))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,K4)),groups3539618377306564664at_int(G3,K4))) ) ).

% sum_mono
tff(fact_8056_sum_Oswap__restrict,axiom,
    ! [A3: set_bool,B2: set_nat,G3: fun_bool_fun_nat_nat,R: fun_bo3288154511880838028t_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups5768501273728744877ol_nat(aa_fun7928520491197679664ol_nat(aa_fun5318524179215677308ol_nat(aTP_Lamm_ks(B2),G3),R),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun4984880352635657191at_nat(aa_fun8028557108643889at_nat(aTP_Lamm_ku(A3),G3),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_8057_sum_Oswap__restrict,axiom,
    ! [A3: set_real,B2: set_nat,G3: fun_real_fun_nat_nat,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups1935376822645274424al_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_kw(B2),G3),R),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_ky(A3),G3),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_8058_sum_Oswap__restrict,axiom,
    ! [A3: set_int,B2: set_nat,G3: fun_int_fun_nat_nat,R: fun_int_fun_nat_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups4541462559716669496nt_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_la(B2),G3),R),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_ld(A3),G3),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_8059_sum_Oswap__restrict,axiom,
    ! [A3: set_complex,B2: set_nat,G3: fun_co636942507509095880at_nat,R: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups5693394587270226106ex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_lf(B2),G3),R),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_li(A3),G3),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_8060_sum_Oswap__restrict,axiom,
    ! [A3: set_Extended_enat,B2: set_nat,G3: fun_Ex7123704652682714774at_nat,R: fun_Ex7682446853002821117t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups2027974829824023292at_nat(aa_fun6947057244751970126at_nat(aa_fun635179117725900299at_nat(aTP_Lamm_lk(B2),G3),R),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun1897775202592251224at_nat(aa_fun594951741515554929at_nat(aTP_Lamm_ln(A3),G3),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_8061_sum_Oswap__restrict,axiom,
    ! [A3: set_bool,B2: set_complex,G3: fun_bo6990880009230973955omplex,R: fun_bo8977660800532043054x_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups2959343415743557771omplex(aa_fun7507808871734033644omplex(aa_fun3415237024035613564omplex(aTP_Lamm_lp(B2),G3),R),A3) = groups7754918857620584856omplex(aa_fun8239209723672002629omplex(aa_fun6558688929985837387omplex(aTP_Lamm_lr(A3),G3),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_8062_sum_Oswap__restrict,axiom,
    ! [A3: set_real,B2: set_complex,G3: fun_re7208826350674946562omplex,R: fun_re3010880487650781423x_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups5754745047067104278omplex(aa_fun4873726854752153866omplex(aa_fun883667864959905433omplex(aTP_Lamm_lt(B2),G3),R),A3) = groups7754918857620584856omplex(aa_fun8128191506076533516omplex(aa_fun840110154952133019omplex(aTP_Lamm_lv(A3),G3),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_8063_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B2: set_complex,G3: fun_na1591161823130732966omplex,R: fun_na5941084329111710283x_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2700696402887535882omplex(aa_fun8019143150332516725omplex(aTP_Lamm_lx(B2),G3),R)),A3) = groups7754918857620584856omplex(aa_fun1613615970879741416omplex(aa_fun7586814990830484307omplex(aTP_Lamm_ma(A3),G3),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_8064_sum_Oswap__restrict,axiom,
    ! [A3: set_int,B2: set_complex,G3: fun_in1307110748760623106omplex,R: fun_in5750378413879446255x_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups3049146728041665814omplex(aa_fun751309451324587018omplex(aa_fun3438240386181387033omplex(aTP_Lamm_mc(B2),G3),R),A3) = groups7754918857620584856omplex(aa_fun2686559102040213132omplex(aa_fun5422379709423809691omplex(aTP_Lamm_mf(A3),G3),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_8065_sum_Oswap__restrict,axiom,
    ! [A3: set_Extended_enat,B2: set_complex,G3: fun_Ex6753287174829011282omplex,R: fun_Ex650070010719229343x_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups6818542070133387226omplex(aa_fun5203025040472162314omplex(aa_fun6421059515561147915omplex(aTP_Lamm_mh(B2),G3),R),A3) = groups7754918857620584856omplex(aa_fun2004122418815753142omplex(aa_fun1910344193496359307omplex(aTP_Lamm_mk(A3),G3),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_8066_sums__If__finite__set,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_hj(A3),F)),groups3539618377306564664at_int(F,A3))) ) ).

% sums_If_finite_set
tff(fact_8067_sums__If__finite__set,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hi(A3),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3))) ) ).

% sums_If_finite_set
tff(fact_8068_sums__If__finite__set,axiom,
    ! [A3: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_hh(A3),F)),aa_set_nat_real(groups6591440286371151544t_real(F),A3))) ) ).

% sums_If_finite_set
tff(fact_8069_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_hm(P),F)),groups3539618377306564664at_int(F,collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_8070_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hl(P),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_8071_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_hk(P),F)),aa_set_nat_real(groups6591440286371151544t_real(F),collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_8072_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => pp(aa_int_bool(sums_int(F),groups3539618377306564664at_int(F,N4))) ) ) ).

% sums_finite
tff(fact_8073_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => pp(aa_nat_bool(sums_nat(F),aa_set_nat_nat(groups3542108847815614940at_nat(F),N4))) ) ) ).

% sums_finite
tff(fact_8074_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => pp(aa_real_bool(sums_real(F),aa_set_nat_real(groups6591440286371151544t_real(F),N4))) ) ) ).

% sums_finite
tff(fact_8075_norm__sum,axiom,
    ! [F: fun_nat_complex,A3: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),A3))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ht(F)),A3))) ).

% norm_sum
tff(fact_8076_norm__sum,axiom,
    ! [F: fun_complex_complex,A3: set_complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,A3))),groups5808333547571424918x_real(aTP_Lamm_ml(F),A3))) ).

% norm_sum
tff(fact_8077_norm__sum,axiom,
    ! [F: fun_nat_real,A3: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),A3))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hw(F)),A3))) ).

% norm_sum
tff(fact_8078_sum__nonpos,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X5)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A3)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_8079_sum__nonpos,axiom,
    ! [A3: set_bool,F: fun_bool_real] :
      ( ! [X5: bool] :
          ( pp(aa_set_bool_bool(member_bool(X5),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_bool_real(F,X5)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups502887708668150281l_real(F,A3)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_8080_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X5)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,A3)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_8081_sum__nonpos,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,X5)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A3)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_8082_sum__nonpos,axiom,
    ! [A3: set_bool,F: fun_bool_rat] :
      ( ! [X5: bool] :
          ( pp(aa_set_bool_bool(member_bool(X5),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_bool_rat(F,X5)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5133371213642249141ol_rat(F,A3)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_8083_sum__nonpos,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X5)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,A3)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_8084_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X5)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A3)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_8085_sum__nonpos,axiom,
    ! [A3: set_real,F: fun_real_nat] :
      ( ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X5)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,A3)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_8086_sum__nonpos,axiom,
    ! [A3: set_bool,F: fun_bool_nat] :
      ( ! [X5: bool] :
          ( pp(aa_set_bool_bool(member_bool(X5),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(F,X5)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups5768501273728744877ol_nat(F,A3)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_8087_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X5)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A3)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_8088_sum__nonneg,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X5))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A3))) ) ).

% sum_nonneg
tff(fact_8089_sum__nonneg,axiom,
    ! [A3: set_bool,F: fun_bool_real] :
      ( ! [X5: bool] :
          ( pp(aa_set_bool_bool(member_bool(X5),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_bool_real(F,X5))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups502887708668150281l_real(F,A3))) ) ).

% sum_nonneg
tff(fact_8090_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X5))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A3))) ) ).

% sum_nonneg
tff(fact_8091_sum__nonneg,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X5))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A3))) ) ).

% sum_nonneg
tff(fact_8092_sum__nonneg,axiom,
    ! [A3: set_bool,F: fun_bool_rat] :
      ( ! [X5: bool] :
          ( pp(aa_set_bool_bool(member_bool(X5),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_bool_rat(F,X5))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups5133371213642249141ol_rat(F,A3))) ) ).

% sum_nonneg
tff(fact_8093_sum__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X5))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A3))) ) ).

% sum_nonneg
tff(fact_8094_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X5))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A3))) ) ).

% sum_nonneg
tff(fact_8095_sum__nonneg,axiom,
    ! [A3: set_real,F: fun_real_nat] :
      ( ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X5))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A3))) ) ).

% sum_nonneg
tff(fact_8096_sum__nonneg,axiom,
    ! [A3: set_bool,F: fun_bool_nat] :
      ( ! [X5: bool] :
          ( pp(aa_set_bool_bool(member_bool(X5),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_bool_nat(F,X5))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups5768501273728744877ol_nat(F,A3))) ) ).

% sum_nonneg
tff(fact_8097_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X5))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A3))) ) ).

% sum_nonneg
tff(fact_8098_sum__mono__inv,axiom,
    ! [F: fun_real_rat,I5: set_real,G3: fun_real_rat,I: real] :
      ( ( groups1300246762558778688al_rat(F,I5) = groups1300246762558778688al_rat(G3,I5) )
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G3,I2))) )
       => ( pp(aa_set_real_bool(member_real(I),I5))
         => ( pp(aa_set_real_bool(finite_finite_real,I5))
           => ( aa_real_rat(F,I) = aa_real_rat(G3,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8099_sum__mono__inv,axiom,
    ! [F: fun_bool_rat,I5: set_bool,G3: fun_bool_rat,I: bool] :
      ( ( groups5133371213642249141ol_rat(F,I5) = groups5133371213642249141ol_rat(G3,I5) )
     => ( ! [I2: bool] :
            ( pp(aa_set_bool_bool(member_bool(I2),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_bool_rat(F,I2)),aa_bool_rat(G3,I2))) )
       => ( pp(aa_set_bool_bool(member_bool(I),I5))
         => ( pp(aa_set_bool_bool(finite_finite_bool,I5))
           => ( aa_bool_rat(F,I) = aa_bool_rat(G3,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8100_sum__mono__inv,axiom,
    ! [F: fun_nat_rat,I5: set_nat,G3: fun_nat_rat,I: nat] :
      ( ( groups2906978787729119204at_rat(F,I5) = groups2906978787729119204at_rat(G3,I5) )
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G3,I2))) )
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
           => ( aa_nat_rat(F,I) = aa_nat_rat(G3,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8101_sum__mono__inv,axiom,
    ! [F: fun_int_rat,I5: set_int,G3: fun_int_rat,I: int] :
      ( ( groups3906332499630173760nt_rat(F,I5) = groups3906332499630173760nt_rat(G3,I5) )
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G3,I2))) )
       => ( pp(aa_set_int_bool(member_int(I),I5))
         => ( pp(aa_set_int_bool(finite_finite_int,I5))
           => ( aa_int_rat(F,I) = aa_int_rat(G3,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8102_sum__mono__inv,axiom,
    ! [F: fun_complex_rat,I5: set_complex,G3: fun_complex_rat,I: complex] :
      ( ( groups5058264527183730370ex_rat(F,I5) = groups5058264527183730370ex_rat(G3,I5) )
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G3,I2))) )
       => ( pp(aa_set_complex_bool(member_complex(I),I5))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
           => ( aa_complex_rat(F,I) = aa_complex_rat(G3,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8103_sum__mono__inv,axiom,
    ! [F: fun_Ex4152069839811072081at_rat,I5: set_Extended_enat,G3: fun_Ex4152069839811072081at_rat,I: extended_enat] :
      ( ( groups1392844769737527556at_rat(F,I5) = groups1392844769737527556at_rat(G3,I5) )
     => ( ! [I2: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Extended_enat_rat(F,I2)),aa_Extended_enat_rat(G3,I2))) )
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I5))
         => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
           => ( aa_Extended_enat_rat(F,I) = aa_Extended_enat_rat(G3,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8104_sum__mono__inv,axiom,
    ! [F: fun_real_nat,I5: set_real,G3: fun_real_nat,I: real] :
      ( ( groups1935376822645274424al_nat(F,I5) = groups1935376822645274424al_nat(G3,I5) )
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G3,I2))) )
       => ( pp(aa_set_real_bool(member_real(I),I5))
         => ( pp(aa_set_real_bool(finite_finite_real,I5))
           => ( aa_real_nat(F,I) = aa_real_nat(G3,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8105_sum__mono__inv,axiom,
    ! [F: fun_bool_nat,I5: set_bool,G3: fun_bool_nat,I: bool] :
      ( ( groups5768501273728744877ol_nat(F,I5) = groups5768501273728744877ol_nat(G3,I5) )
     => ( ! [I2: bool] :
            ( pp(aa_set_bool_bool(member_bool(I2),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(F,I2)),aa_bool_nat(G3,I2))) )
       => ( pp(aa_set_bool_bool(member_bool(I),I5))
         => ( pp(aa_set_bool_bool(finite_finite_bool,I5))
           => ( aa_bool_nat(F,I) = aa_bool_nat(G3,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8106_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I5: set_int,G3: fun_int_nat,I: int] :
      ( ( groups4541462559716669496nt_nat(F,I5) = groups4541462559716669496nt_nat(G3,I5) )
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G3,I2))) )
       => ( pp(aa_set_int_bool(member_int(I),I5))
         => ( pp(aa_set_int_bool(finite_finite_int,I5))
           => ( aa_int_nat(F,I) = aa_int_nat(G3,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8107_sum__mono__inv,axiom,
    ! [F: fun_complex_nat,I5: set_complex,G3: fun_complex_nat,I: complex] :
      ( ( groups5693394587270226106ex_nat(F,I5) = groups5693394587270226106ex_nat(G3,I5) )
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G3,I2))) )
       => ( pp(aa_set_complex_bool(member_complex(I),I5))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
           => ( aa_complex_nat(F,I) = aa_complex_nat(G3,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8108_sum__cong__Suc,axiom,
    ! [A3: set_nat,F: fun_nat_nat,G3: fun_nat_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3))
     => ( ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X5)),A3))
           => ( aa_nat_nat(F,aa_nat_nat(suc,X5)) = aa_nat_nat(G3,aa_nat_nat(suc,X5)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(G3),A3) ) ) ) ).

% sum_cong_Suc
tff(fact_8109_sum__cong__Suc,axiom,
    ! [A3: set_nat,F: fun_nat_real,G3: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3))
     => ( ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X5)),A3))
           => ( aa_nat_real(F,aa_nat_nat(suc,X5)) = aa_nat_real(G3,aa_nat_nat(suc,X5)) ) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),A3) = aa_set_nat_real(groups6591440286371151544t_real(G3),A3) ) ) ) ).

% sum_cong_Suc
tff(fact_8110_sums__single,axiom,
    ! [I: nat,F: fun_nat_real] : pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ha(I),F)),aa_nat_real(F,I))) ).

% sums_single
tff(fact_8111_sums__single,axiom,
    ! [I: nat,F: fun_nat_nat] : pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hb(I),F)),aa_nat_nat(F,I))) ).

% sums_single
tff(fact_8112_sums__single,axiom,
    ! [I: nat,F: fun_nat_int] : pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_hc(I),F)),aa_nat_int(F,I))) ).

% sums_single
tff(fact_8113_sum_Ointer__filter,axiom,
    ! [A3: set_bool,G3: fun_bool_real,P: fun_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( groups502887708668150281l_real(G3,collect_bool(aa_fun8098540714523660868l_bool(aTP_Lamm_fd(A3),P))) = groups502887708668150281l_real(aa_fun316222562968675029l_real(aTP_Lamm_mm(G3),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8114_sum_Ointer__filter,axiom,
    ! [A3: set_real,G3: fun_real_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups8097168146408367636l_real(G3,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_fe(A3),P))) = groups8097168146408367636l_real(aa_fun6348708079608492509l_real(aTP_Lamm_mn(G3),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8115_sum_Ointer__filter,axiom,
    ! [A3: set_int,G3: fun_int_real,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups8778361861064173332t_real(G3,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_fi(A3),P))) = groups8778361861064173332t_real(aa_fun6974987029372589917t_real(aTP_Lamm_mo(G3),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8116_sum_Ointer__filter,axiom,
    ! [A3: set_complex,G3: fun_complex_real,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5808333547571424918x_real(G3,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_mp(A3),P))) = groups5808333547571424918x_real(aa_fun2902006202505214429x_real(aTP_Lamm_mq(G3),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8117_sum_Ointer__filter,axiom,
    ! [A3: set_Extended_enat,G3: fun_Ex1451620480498224165t_real,P: fun_Ex8563670519792538868t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups4148127829035722712t_real(G3,collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_mr(A3),P))) = groups4148127829035722712t_real(aa_fun1090353380788292083t_real(aTP_Lamm_ms(G3),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8118_sum_Ointer__filter,axiom,
    ! [A3: set_bool,G3: fun_bool_rat,P: fun_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( groups5133371213642249141ol_rat(G3,collect_bool(aa_fun8098540714523660868l_bool(aTP_Lamm_fd(A3),P))) = groups5133371213642249141ol_rat(aa_fun4278745445771542273ol_rat(aTP_Lamm_mt(G3),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8119_sum_Ointer__filter,axiom,
    ! [A3: set_real,G3: fun_real_rat,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups1300246762558778688al_rat(G3,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_fe(A3),P))) = groups1300246762558778688al_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_mu(G3),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8120_sum_Ointer__filter,axiom,
    ! [A3: set_nat,G3: fun_nat_rat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( groups2906978787729119204at_rat(G3,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_fh(A3),P))) = groups2906978787729119204at_rat(aa_fun5656162108953708681at_rat(aTP_Lamm_mv(G3),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8121_sum_Ointer__filter,axiom,
    ! [A3: set_int,G3: fun_int_rat,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups3906332499630173760nt_rat(G3,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_fi(A3),P))) = groups3906332499630173760nt_rat(aa_fun4400602662665189257nt_rat(aTP_Lamm_mw(G3),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8122_sum_Ointer__filter,axiom,
    ! [A3: set_complex,G3: fun_complex_rat,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5058264527183730370ex_rat(G3,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_mp(A3),P))) = groups5058264527183730370ex_rat(aa_fun5529999419959482889ex_rat(aTP_Lamm_mx(G3),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8123_sums__If__finite__set_H,axiom,
    ! [G3: fun_nat_real,S: real,A3: set_nat,S5: real,F: fun_nat_real] :
      ( pp(aa_real_bool(sums_real(G3),S))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A3))
       => ( ( S5 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S),aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_my(G3),F)),A3)) )
         => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_mz(G3),A3),F)),S5)) ) ) ) ).

% sums_If_finite_set'
tff(fact_8124_sum__le__included,axiom,
    ! [S2: set_int,T: set_int,G3: fun_int_real,I: fun_int_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G3,X5))) )
         => ( ! [X5: int] :
                ( pp(aa_set_int_bool(member_int(X5),S2))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_int(I,Xa) = X5 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X5)),aa_int_real(G3,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S2)),groups8778361861064173332t_real(G3,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8125_sum__le__included,axiom,
    ! [S2: set_int,T: set_complex,G3: fun_complex_real,I: fun_complex_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G3,X5))) )
         => ( ! [X5: int] :
                ( pp(aa_set_int_bool(member_int(X5),S2))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_int(I,Xa) = X5 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X5)),aa_complex_real(G3,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S2)),groups5808333547571424918x_real(G3,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8126_sum__le__included,axiom,
    ! [S2: set_int,T: set_Extended_enat,G3: fun_Ex1451620480498224165t_real,I: fun_Ex3482447378695272101at_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(G3,X5))) )
         => ( ! [X5: int] :
                ( pp(aa_set_int_bool(member_int(X5),S2))
               => ? [Xa: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),T))
                    & ( aa_Extended_enat_int(I,Xa) = X5 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X5)),aa_Ext2541572767299617110t_real(G3,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S2)),groups4148127829035722712t_real(G3,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8127_sum__le__included,axiom,
    ! [S2: set_complex,T: set_int,G3: fun_int_real,I: fun_int_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G3,X5))) )
         => ( ! [X5: complex] :
                ( pp(aa_set_complex_bool(member_complex(X5),S2))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_complex(I,Xa) = X5 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X5)),aa_int_real(G3,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S2)),groups8778361861064173332t_real(G3,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8128_sum__le__included,axiom,
    ! [S2: set_complex,T: set_complex,G3: fun_complex_real,I: fun_complex_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G3,X5))) )
         => ( ! [X5: complex] :
                ( pp(aa_set_complex_bool(member_complex(X5),S2))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_complex(I,Xa) = X5 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X5)),aa_complex_real(G3,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S2)),groups5808333547571424918x_real(G3,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8129_sum__le__included,axiom,
    ! [S2: set_complex,T: set_Extended_enat,G3: fun_Ex1451620480498224165t_real,I: fun_Ex3741802288643809319omplex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(G3,X5))) )
         => ( ! [X5: complex] :
                ( pp(aa_set_complex_bool(member_complex(X5),S2))
               => ? [Xa: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),T))
                    & ( aa_Ext8748583412541189208omplex(I,Xa) = X5 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X5)),aa_Ext2541572767299617110t_real(G3,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S2)),groups4148127829035722712t_real(G3,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8130_sum__le__included,axiom,
    ! [S2: set_Extended_enat,T: set_int,G3: fun_int_real,I: fun_in5914218760682845459d_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G3,X5))) )
         => ( ! [X5: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),S2))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_Extended_enat(I,Xa) = X5 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,X5)),aa_int_real(G3,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups4148127829035722712t_real(F,S2)),groups8778361861064173332t_real(G3,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8131_sum__le__included,axiom,
    ! [S2: set_Extended_enat,T: set_complex,G3: fun_complex_real,I: fun_co6496365337113880337d_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G3,X5))) )
         => ( ! [X5: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),S2))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_com3683005661447327866d_enat(I,Xa) = X5 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,X5)),aa_complex_real(G3,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups4148127829035722712t_real(F,S2)),groups5808333547571424918x_real(G3,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8132_sum__le__included,axiom,
    ! [S2: set_Extended_enat,T: set_Extended_enat,G3: fun_Ex1451620480498224165t_real,I: fun_Ex5852502975177633731d_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(G3,X5))) )
         => ( ! [X5: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),S2))
               => ? [Xa: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),T))
                    & ( aa_Ext7969882492699152700d_enat(I,Xa) = X5 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,X5)),aa_Ext2541572767299617110t_real(G3,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups4148127829035722712t_real(F,S2)),groups4148127829035722712t_real(G3,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8133_sum__le__included,axiom,
    ! [S2: set_nat,T: set_nat,G3: fun_nat_rat,I: fun_nat_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T))
       => ( ! [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G3,X5))) )
         => ( ! [X5: nat] :
                ( pp(aa_set_nat_bool(member_nat(X5),S2))
               => ? [Xa: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa),T))
                    & ( aa_nat_nat(I,Xa) = X5 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X5)),aa_nat_rat(G3,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,S2)),groups2906978787729119204at_rat(G3,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8134_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ! [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X5))) )
       => ( ( groups8097168146408367636l_real(F,A3) = zero_zero_real )
        <=> ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),A3))
             => ( aa_real_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8135_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_bool,F: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( ! [X5: bool] :
            ( pp(aa_set_bool_bool(member_bool(X5),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_bool_real(F,X5))) )
       => ( ( groups502887708668150281l_real(F,A3) = zero_zero_real )
        <=> ! [X3: bool] :
              ( pp(aa_set_bool_bool(member_bool(X3),A3))
             => ( aa_bool_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8136_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X5))) )
       => ( ( groups8778361861064173332t_real(F,A3) = zero_zero_real )
        <=> ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A3))
             => ( aa_int_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8137_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ! [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X5))) )
       => ( ( groups5808333547571424918x_real(F,A3) = zero_zero_real )
        <=> ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),A3))
             => ( aa_complex_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8138_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ! [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,X5))) )
       => ( ( groups4148127829035722712t_real(F,A3) = zero_zero_real )
        <=> ! [X3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A3))
             => ( aa_Ext2541572767299617110t_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8139_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ! [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X5))) )
       => ( ( groups1300246762558778688al_rat(F,A3) = zero_zero_rat )
        <=> ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),A3))
             => ( aa_real_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8140_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_bool,F: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( ! [X5: bool] :
            ( pp(aa_set_bool_bool(member_bool(X5),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_bool_rat(F,X5))) )
       => ( ( groups5133371213642249141ol_rat(F,A3) = zero_zero_rat )
        <=> ! [X3: bool] :
              ( pp(aa_set_bool_bool(member_bool(X3),A3))
             => ( aa_bool_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8141_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X5))) )
       => ( ( groups2906978787729119204at_rat(F,A3) = zero_zero_rat )
        <=> ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),A3))
             => ( aa_nat_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8142_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X5))) )
       => ( ( groups3906332499630173760nt_rat(F,A3) = zero_zero_rat )
        <=> ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A3))
             => ( aa_int_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8143_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ! [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X5))) )
       => ( ( groups5058264527183730370ex_rat(F,A3) = zero_zero_rat )
        <=> ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),A3))
             => ( aa_complex_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8144_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: fun_int_real,G3: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X5)),aa_int_real(G3,X5))) )
       => ( ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(G3,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A3)),groups8778361861064173332t_real(G3,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8145_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,F: fun_complex_real,G3: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ! [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X5)),aa_complex_real(G3,X5))) )
       => ( ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X4)),aa_complex_real(G3,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A3)),groups5808333547571424918x_real(G3,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8146_sum__strict__mono__ex1,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real,G3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ! [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,X5)),aa_Ext2541572767299617110t_real(G3,X5))) )
       => ( ? [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(G3,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups4148127829035722712t_real(F,A3)),groups4148127829035722712t_real(G3,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8147_sum__strict__mono__ex1,axiom,
    ! [A3: set_nat,F: fun_nat_rat,G3: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X5)),aa_nat_rat(G3,X5))) )
       => ( ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A3))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(G3,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A3)),groups2906978787729119204at_rat(G3,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8148_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: fun_int_rat,G3: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X5)),aa_int_rat(G3,X5))) )
       => ( ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A3))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(G3,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(G3,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8149_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,F: fun_complex_rat,G3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ! [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X5)),aa_complex_rat(G3,X5))) )
       => ( ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A3))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X4)),aa_complex_rat(G3,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(G3,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8150_sum__strict__mono__ex1,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat,G3: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ! [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Extended_enat_rat(F,X5)),aa_Extended_enat_rat(G3,X5))) )
       => ( ? [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A3))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Extended_enat_rat(F,X4)),aa_Extended_enat_rat(G3,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1392844769737527556at_rat(F,A3)),groups1392844769737527556at_rat(G3,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8151_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: fun_int_nat,G3: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X5)),aa_int_nat(G3,X5))) )
       => ( ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A3))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X4)),aa_int_nat(G3,X4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(G3,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8152_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,F: fun_complex_nat,G3: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ! [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,X5)),aa_complex_nat(G3,X5))) )
       => ( ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A3))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,X4)),aa_complex_nat(G3,X4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A3)),groups5693394587270226106ex_nat(G3,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8153_sum__strict__mono__ex1,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex7660298398204468809at_nat,G3: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ! [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Extended_enat_nat(F,X5)),aa_Extended_enat_nat(G3,X5))) )
       => ( ? [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A3))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(G3,X4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups2027974829824023292at_nat(F,A3)),groups2027974829824023292at_nat(G3,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8154_sum_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S: set_int,H: fun_int_real,G3: fun_int_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,zero_zero_real),zero_zero_real))
     => ( ! [X1: real,Y1: real,X22: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X22))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X22),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S))
         => ( ! [X5: int] :
                ( pp(aa_set_int_bool(member_int(X5),S))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_int_real(H,X5)),aa_int_real(G3,X5))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups8778361861064173332t_real(H,S)),groups8778361861064173332t_real(G3,S))) ) ) ) ) ).

% sum.related
tff(fact_8155_sum_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S: set_complex,H: fun_complex_real,G3: fun_complex_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,zero_zero_real),zero_zero_real))
     => ( ! [X1: real,Y1: real,X22: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X22))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X22),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
         => ( ! [X5: complex] :
                ( pp(aa_set_complex_bool(member_complex(X5),S))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_complex_real(H,X5)),aa_complex_real(G3,X5))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups5808333547571424918x_real(H,S)),groups5808333547571424918x_real(G3,S))) ) ) ) ) ).

% sum.related
tff(fact_8156_sum_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S: set_Extended_enat,H: fun_Ex1451620480498224165t_real,G3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,zero_zero_real),zero_zero_real))
     => ( ! [X1: real,Y1: real,X22: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X22))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X22),Y23))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
         => ( ! [X5: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),S))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_Ext2541572767299617110t_real(H,X5)),aa_Ext2541572767299617110t_real(G3,X5))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups4148127829035722712t_real(H,S)),groups4148127829035722712t_real(G3,S))) ) ) ) ) ).

% sum.related
tff(fact_8157_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S: set_nat,H: fun_nat_rat,G3: fun_nat_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,zero_zero_rat),zero_zero_rat))
     => ( ! [X1: rat,Y1: rat,X22: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X1),X22))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X1),Y1)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X22),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S))
         => ( ! [X5: nat] :
                ( pp(aa_set_nat_bool(member_nat(X5),S))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_nat_rat(H,X5)),aa_nat_rat(G3,X5))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups2906978787729119204at_rat(H,S)),groups2906978787729119204at_rat(G3,S))) ) ) ) ) ).

% sum.related
tff(fact_8158_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S: set_int,H: fun_int_rat,G3: fun_int_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,zero_zero_rat),zero_zero_rat))
     => ( ! [X1: rat,Y1: rat,X22: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X1),X22))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X1),Y1)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X22),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S))
         => ( ! [X5: int] :
                ( pp(aa_set_int_bool(member_int(X5),S))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_int_rat(H,X5)),aa_int_rat(G3,X5))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups3906332499630173760nt_rat(H,S)),groups3906332499630173760nt_rat(G3,S))) ) ) ) ) ).

% sum.related
tff(fact_8159_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S: set_complex,H: fun_complex_rat,G3: fun_complex_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,zero_zero_rat),zero_zero_rat))
     => ( ! [X1: rat,Y1: rat,X22: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X1),X22))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X1),Y1)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X22),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
         => ( ! [X5: complex] :
                ( pp(aa_set_complex_bool(member_complex(X5),S))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_complex_rat(H,X5)),aa_complex_rat(G3,X5))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups5058264527183730370ex_rat(H,S)),groups5058264527183730370ex_rat(G3,S))) ) ) ) ) ).

% sum.related
tff(fact_8160_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S: set_Extended_enat,H: fun_Ex4152069839811072081at_rat,G3: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,zero_zero_rat),zero_zero_rat))
     => ( ! [X1: rat,Y1: rat,X22: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X1),X22))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X1),Y1)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X22),Y23))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
         => ( ! [X5: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),S))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_Extended_enat_rat(H,X5)),aa_Extended_enat_rat(G3,X5))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups1392844769737527556at_rat(H,S)),groups1392844769737527556at_rat(G3,S))) ) ) ) ) ).

% sum.related
tff(fact_8161_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S: set_int,H: fun_int_nat,G3: fun_int_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,zero_zero_nat),zero_zero_nat))
     => ( ! [X1: nat,Y1: nat,X22: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X1),X22))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y1),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S))
         => ( ! [X5: int] :
                ( pp(aa_set_int_bool(member_int(X5),S))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_int_nat(H,X5)),aa_int_nat(G3,X5))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups4541462559716669496nt_nat(H,S)),groups4541462559716669496nt_nat(G3,S))) ) ) ) ) ).

% sum.related
tff(fact_8162_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S: set_complex,H: fun_complex_nat,G3: fun_complex_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,zero_zero_nat),zero_zero_nat))
     => ( ! [X1: nat,Y1: nat,X22: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X1),X22))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y1),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
         => ( ! [X5: complex] :
                ( pp(aa_set_complex_bool(member_complex(X5),S))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_complex_nat(H,X5)),aa_complex_nat(G3,X5))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups5693394587270226106ex_nat(H,S)),groups5693394587270226106ex_nat(G3,S))) ) ) ) ) ).

% sum.related
tff(fact_8163_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S: set_Extended_enat,H: fun_Ex7660298398204468809at_nat,G3: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,zero_zero_nat),zero_zero_nat))
     => ( ! [X1: nat,Y1: nat,X22: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X1),X22))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y1),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y23))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
         => ( ! [X5: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),S))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_Extended_enat_nat(H,X5)),aa_Extended_enat_nat(G3,X5))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups2027974829824023292at_nat(H,S)),groups2027974829824023292at_nat(G3,S))) ) ) ) ) ).

% sum.related
tff(fact_8164_sum__strict__mono,axiom,
    ! [A3: set_complex,F: fun_complex_real,G3: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( A3 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),A3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X5)),aa_complex_real(G3,X5))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A3)),groups5808333547571424918x_real(G3,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_8165_sum__strict__mono,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real,G3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,X5)),aa_Ext2541572767299617110t_real(G3,X5))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups4148127829035722712t_real(F,A3)),groups4148127829035722712t_real(G3,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_8166_sum__strict__mono,axiom,
    ! [A3: set_real,F: fun_real_real,G3: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ( A3 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),A3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X5)),aa_real_real(G3,X5))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A3)),groups8097168146408367636l_real(G3,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_8167_sum__strict__mono,axiom,
    ! [A3: set_bool,F: fun_bool_real,G3: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( ( A3 != bot_bot_set_bool )
       => ( ! [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),A3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_bool_real(F,X5)),aa_bool_real(G3,X5))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups502887708668150281l_real(F,A3)),groups502887708668150281l_real(G3,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_8168_sum__strict__mono,axiom,
    ! [A3: set_int,F: fun_int_real,G3: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( A3 != bot_bot_set_int )
       => ( ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),A3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X5)),aa_int_real(G3,X5))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A3)),groups8778361861064173332t_real(G3,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_8169_sum__strict__mono,axiom,
    ! [A3: set_complex,F: fun_complex_rat,G3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( A3 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),A3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X5)),aa_complex_rat(G3,X5))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(G3,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_8170_sum__strict__mono,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat,G3: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Extended_enat_rat(F,X5)),aa_Extended_enat_rat(G3,X5))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1392844769737527556at_rat(F,A3)),groups1392844769737527556at_rat(G3,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_8171_sum__strict__mono,axiom,
    ! [A3: set_real,F: fun_real_rat,G3: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ( A3 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),A3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X5)),aa_real_rat(G3,X5))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1300246762558778688al_rat(F,A3)),groups1300246762558778688al_rat(G3,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_8172_sum__strict__mono,axiom,
    ! [A3: set_bool,F: fun_bool_rat,G3: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( ( A3 != bot_bot_set_bool )
       => ( ! [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),A3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_bool_rat(F,X5)),aa_bool_rat(G3,X5))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5133371213642249141ol_rat(F,A3)),groups5133371213642249141ol_rat(G3,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_8173_sum__strict__mono,axiom,
    ! [A3: set_nat,F: fun_nat_rat,G3: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),A3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X5)),aa_nat_rat(G3,X5))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A3)),groups2906978787729119204at_rat(G3,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_8174_sum_Oinsert__if,axiom,
    ! [A3: set_real,X: real,G3: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ( pp(aa_set_real_bool(member_real(X),A3))
         => ( groups8097168146408367636l_real(G3,insert_real(X,A3)) = groups8097168146408367636l_real(G3,A3) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A3))
         => ( groups8097168146408367636l_real(G3,insert_real(X,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G3,X)),groups8097168146408367636l_real(G3,A3)) ) ) ) ) ).

% sum.insert_if
tff(fact_8175_sum_Oinsert__if,axiom,
    ! [A3: set_bool,X: bool,G3: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( ( pp(aa_set_bool_bool(member_bool(X),A3))
         => ( groups502887708668150281l_real(G3,insert_bool(X,A3)) = groups502887708668150281l_real(G3,A3) ) )
        & ( ~ pp(aa_set_bool_bool(member_bool(X),A3))
         => ( groups502887708668150281l_real(G3,insert_bool(X,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_bool_real(G3,X)),groups502887708668150281l_real(G3,A3)) ) ) ) ) ).

% sum.insert_if
tff(fact_8176_sum_Oinsert__if,axiom,
    ! [A3: set_int,X: int,G3: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( pp(aa_set_int_bool(member_int(X),A3))
         => ( groups8778361861064173332t_real(G3,insert_int(X,A3)) = groups8778361861064173332t_real(G3,A3) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A3))
         => ( groups8778361861064173332t_real(G3,insert_int(X,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G3,X)),groups8778361861064173332t_real(G3,A3)) ) ) ) ) ).

% sum.insert_if
tff(fact_8177_sum_Oinsert__if,axiom,
    ! [A3: set_complex,X: complex,G3: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A3))
         => ( groups5808333547571424918x_real(G3,insert_complex(X,A3)) = groups5808333547571424918x_real(G3,A3) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A3))
         => ( groups5808333547571424918x_real(G3,insert_complex(X,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G3,X)),groups5808333547571424918x_real(G3,A3)) ) ) ) ) ).

% sum.insert_if
tff(fact_8178_sum_Oinsert__if,axiom,
    ! [A3: set_Extended_enat,X: extended_enat,G3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A3))
         => ( groups4148127829035722712t_real(G3,insert_Extended_enat(X,A3)) = groups4148127829035722712t_real(G3,A3) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A3))
         => ( groups4148127829035722712t_real(G3,insert_Extended_enat(X,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Ext2541572767299617110t_real(G3,X)),groups4148127829035722712t_real(G3,A3)) ) ) ) ) ).

% sum.insert_if
tff(fact_8179_sum_Oinsert__if,axiom,
    ! [A3: set_real,X: real,G3: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ( pp(aa_set_real_bool(member_real(X),A3))
         => ( groups1300246762558778688al_rat(G3,insert_real(X,A3)) = groups1300246762558778688al_rat(G3,A3) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A3))
         => ( groups1300246762558778688al_rat(G3,insert_real(X,A3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(G3,X)),groups1300246762558778688al_rat(G3,A3)) ) ) ) ) ).

% sum.insert_if
tff(fact_8180_sum_Oinsert__if,axiom,
    ! [A3: set_bool,X: bool,G3: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( ( pp(aa_set_bool_bool(member_bool(X),A3))
         => ( groups5133371213642249141ol_rat(G3,insert_bool(X,A3)) = groups5133371213642249141ol_rat(G3,A3) ) )
        & ( ~ pp(aa_set_bool_bool(member_bool(X),A3))
         => ( groups5133371213642249141ol_rat(G3,insert_bool(X,A3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_bool_rat(G3,X)),groups5133371213642249141ol_rat(G3,A3)) ) ) ) ) ).

% sum.insert_if
tff(fact_8181_sum_Oinsert__if,axiom,
    ! [A3: set_nat,X: nat,G3: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( pp(aa_set_nat_bool(member_nat(X),A3))
         => ( groups2906978787729119204at_rat(G3,insert_nat(X,A3)) = groups2906978787729119204at_rat(G3,A3) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A3))
         => ( groups2906978787729119204at_rat(G3,insert_nat(X,A3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G3,X)),groups2906978787729119204at_rat(G3,A3)) ) ) ) ) ).

% sum.insert_if
tff(fact_8182_sum_Oinsert__if,axiom,
    ! [A3: set_int,X: int,G3: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( pp(aa_set_int_bool(member_int(X),A3))
         => ( groups3906332499630173760nt_rat(G3,insert_int(X,A3)) = groups3906332499630173760nt_rat(G3,A3) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A3))
         => ( groups3906332499630173760nt_rat(G3,insert_int(X,A3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(G3,X)),groups3906332499630173760nt_rat(G3,A3)) ) ) ) ) ).

% sum.insert_if
tff(fact_8183_sum_Oinsert__if,axiom,
    ! [A3: set_complex,X: complex,G3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A3))
         => ( groups5058264527183730370ex_rat(G3,insert_complex(X,A3)) = groups5058264527183730370ex_rat(G3,A3) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A3))
         => ( groups5058264527183730370ex_rat(G3,insert_complex(X,A3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(G3,X)),groups5058264527183730370ex_rat(G3,A3)) ) ) ) ) ).

% sum.insert_if
tff(fact_8184_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_real,S: set_real,I: fun_real_real,J: fun_real_real,T2: set_real,G3: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T5))
       => ( ! [A: real] :
              ( pp(aa_set_real_bool(member_real(A),minus_minus_set_real(S,S5)))
             => ( aa_real_real(I,aa_real_real(J,A)) = A ) )
         => ( ! [A: real] :
                ( pp(aa_set_real_bool(member_real(A),minus_minus_set_real(S,S5)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J,A)),minus_minus_set_real(T2,T5))) )
           => ( ! [B: real] :
                  ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(T2,T5)))
                 => ( aa_real_real(J,aa_real_real(I,B)) = B ) )
             => ( ! [B: real] :
                    ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(T2,T5)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I,B)),minus_minus_set_real(S,S5))) )
               => ( ! [A: real] :
                      ( pp(aa_set_real_bool(member_real(A),S5))
                     => ( aa_real_real(G3,A) = zero_zero_real ) )
                 => ( ! [B: real] :
                        ( pp(aa_set_real_bool(member_real(B),T5))
                       => ( aa_real_real(H,B) = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( pp(aa_set_real_bool(member_real(A),S))
                         => ( aa_real_real(H,aa_real_real(J,A)) = aa_real_real(G3,A) ) )
                     => ( groups8097168146408367636l_real(G3,S) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8185_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_bool,S: set_real,I: fun_bool_real,J: fun_real_bool,T2: set_bool,G3: fun_real_real,H: fun_bool_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_bool_bool(finite_finite_bool,T5))
       => ( ! [A: real] :
              ( pp(aa_set_real_bool(member_real(A),minus_minus_set_real(S,S5)))
             => ( aa_bool_real(I,aa_real_bool(J,A)) = A ) )
         => ( ! [A: real] :
                ( pp(aa_set_real_bool(member_real(A),minus_minus_set_real(S,S5)))
               => pp(aa_set_bool_bool(member_bool(aa_real_bool(J,A)),minus_minus_set_bool(T2,T5))) )
           => ( ! [B: bool] :
                  ( pp(aa_set_bool_bool(member_bool(B),minus_minus_set_bool(T2,T5)))
                 => ( pp(aa_real_bool(J,aa_bool_real(I,B)))
                  <=> pp(B) ) )
             => ( ! [B: bool] :
                    ( pp(aa_set_bool_bool(member_bool(B),minus_minus_set_bool(T2,T5)))
                   => pp(aa_set_real_bool(member_real(aa_bool_real(I,B)),minus_minus_set_real(S,S5))) )
               => ( ! [A: real] :
                      ( pp(aa_set_real_bool(member_real(A),S5))
                     => ( aa_real_real(G3,A) = zero_zero_real ) )
                 => ( ! [B: bool] :
                        ( pp(aa_set_bool_bool(member_bool(B),T5))
                       => ( aa_bool_real(H,B) = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( pp(aa_set_real_bool(member_real(A),S))
                         => ( aa_bool_real(H,aa_real_bool(J,A)) = aa_real_real(G3,A) ) )
                     => ( groups8097168146408367636l_real(G3,S) = groups502887708668150281l_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8186_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_bool,T5: set_real,S: set_bool,I: fun_real_bool,J: fun_bool_real,T2: set_real,G3: fun_bool_real,H: fun_real_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T5))
       => ( ! [A: bool] :
              ( pp(aa_set_bool_bool(member_bool(A),minus_minus_set_bool(S,S5)))
             => ( pp(aa_real_bool(I,aa_bool_real(J,A)))
              <=> pp(A) ) )
         => ( ! [A: bool] :
                ( pp(aa_set_bool_bool(member_bool(A),minus_minus_set_bool(S,S5)))
               => pp(aa_set_real_bool(member_real(aa_bool_real(J,A)),minus_minus_set_real(T2,T5))) )
           => ( ! [B: real] :
                  ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(T2,T5)))
                 => ( aa_bool_real(J,aa_real_bool(I,B)) = B ) )
             => ( ! [B: real] :
                    ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(T2,T5)))
                   => pp(aa_set_bool_bool(member_bool(aa_real_bool(I,B)),minus_minus_set_bool(S,S5))) )
               => ( ! [A: bool] :
                      ( pp(aa_set_bool_bool(member_bool(A),S5))
                     => ( aa_bool_real(G3,A) = zero_zero_real ) )
                 => ( ! [B: real] :
                        ( pp(aa_set_real_bool(member_real(B),T5))
                       => ( aa_real_real(H,B) = zero_zero_real ) )
                   => ( ! [A: bool] :
                          ( pp(aa_set_bool_bool(member_bool(A),S))
                         => ( aa_real_real(H,aa_bool_real(J,A)) = aa_bool_real(G3,A) ) )
                     => ( groups502887708668150281l_real(G3,S) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8187_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_bool,T5: set_bool,S: set_bool,I: fun_bool_bool,J: fun_bool_bool,T2: set_bool,G3: fun_bool_real,H: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S5))
     => ( pp(aa_set_bool_bool(finite_finite_bool,T5))
       => ( ! [A: bool] :
              ( pp(aa_set_bool_bool(member_bool(A),minus_minus_set_bool(S,S5)))
             => ( pp(aa_bool_bool(I,aa_bool_bool(J,A)))
              <=> pp(A) ) )
         => ( ! [A: bool] :
                ( pp(aa_set_bool_bool(member_bool(A),minus_minus_set_bool(S,S5)))
               => pp(aa_set_bool_bool(member_bool(aa_bool_bool(J,A)),minus_minus_set_bool(T2,T5))) )
           => ( ! [B: bool] :
                  ( pp(aa_set_bool_bool(member_bool(B),minus_minus_set_bool(T2,T5)))
                 => ( pp(aa_bool_bool(J,aa_bool_bool(I,B)))
                  <=> pp(B) ) )
             => ( ! [B: bool] :
                    ( pp(aa_set_bool_bool(member_bool(B),minus_minus_set_bool(T2,T5)))
                   => pp(aa_set_bool_bool(member_bool(aa_bool_bool(I,B)),minus_minus_set_bool(S,S5))) )
               => ( ! [A: bool] :
                      ( pp(aa_set_bool_bool(member_bool(A),S5))
                     => ( aa_bool_real(G3,A) = zero_zero_real ) )
                 => ( ! [B: bool] :
                        ( pp(aa_set_bool_bool(member_bool(B),T5))
                       => ( aa_bool_real(H,B) = zero_zero_real ) )
                   => ( ! [A: bool] :
                          ( pp(aa_set_bool_bool(member_bool(A),S))
                         => ( aa_bool_real(H,aa_bool_bool(J,A)) = aa_bool_real(G3,A) ) )
                     => ( groups502887708668150281l_real(G3,S) = groups502887708668150281l_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8188_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_int,S: set_real,I: fun_int_real,J: fun_real_int,T2: set_int,G3: fun_real_real,H: fun_int_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T5))
       => ( ! [A: real] :
              ( pp(aa_set_real_bool(member_real(A),minus_minus_set_real(S,S5)))
             => ( aa_int_real(I,aa_real_int(J,A)) = A ) )
         => ( ! [A: real] :
                ( pp(aa_set_real_bool(member_real(A),minus_minus_set_real(S,S5)))
               => pp(aa_set_int_bool(member_int(aa_real_int(J,A)),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T2),T5))) )
           => ( ! [B: int] :
                  ( pp(aa_set_int_bool(member_int(B),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T2),T5)))
                 => ( aa_real_int(J,aa_int_real(I,B)) = B ) )
             => ( ! [B: int] :
                    ( pp(aa_set_int_bool(member_int(B),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T2),T5)))
                   => pp(aa_set_real_bool(member_real(aa_int_real(I,B)),minus_minus_set_real(S,S5))) )
               => ( ! [A: real] :
                      ( pp(aa_set_real_bool(member_real(A),S5))
                     => ( aa_real_real(G3,A) = zero_zero_real ) )
                 => ( ! [B: int] :
                        ( pp(aa_set_int_bool(member_int(B),T5))
                       => ( aa_int_real(H,B) = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( pp(aa_set_real_bool(member_real(A),S))
                         => ( aa_int_real(H,aa_real_int(J,A)) = aa_real_real(G3,A) ) )
                     => ( groups8097168146408367636l_real(G3,S) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8189_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_bool,T5: set_int,S: set_bool,I: fun_int_bool,J: fun_bool_int,T2: set_int,G3: fun_bool_real,H: fun_int_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T5))
       => ( ! [A: bool] :
              ( pp(aa_set_bool_bool(member_bool(A),minus_minus_set_bool(S,S5)))
             => ( pp(aa_int_bool(I,aa_bool_int(J,A)))
              <=> pp(A) ) )
         => ( ! [A: bool] :
                ( pp(aa_set_bool_bool(member_bool(A),minus_minus_set_bool(S,S5)))
               => pp(aa_set_int_bool(member_int(aa_bool_int(J,A)),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T2),T5))) )
           => ( ! [B: int] :
                  ( pp(aa_set_int_bool(member_int(B),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T2),T5)))
                 => ( aa_bool_int(J,aa_int_bool(I,B)) = B ) )
             => ( ! [B: int] :
                    ( pp(aa_set_int_bool(member_int(B),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T2),T5)))
                   => pp(aa_set_bool_bool(member_bool(aa_int_bool(I,B)),minus_minus_set_bool(S,S5))) )
               => ( ! [A: bool] :
                      ( pp(aa_set_bool_bool(member_bool(A),S5))
                     => ( aa_bool_real(G3,A) = zero_zero_real ) )
                 => ( ! [B: int] :
                        ( pp(aa_set_int_bool(member_int(B),T5))
                       => ( aa_int_real(H,B) = zero_zero_real ) )
                   => ( ! [A: bool] :
                          ( pp(aa_set_bool_bool(member_bool(A),S))
                         => ( aa_int_real(H,aa_bool_int(J,A)) = aa_bool_real(G3,A) ) )
                     => ( groups502887708668150281l_real(G3,S) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8190_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_complex,S: set_real,I: fun_complex_real,J: fun_real_complex,T2: set_complex,G3: fun_real_real,H: fun_complex_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
       => ( ! [A: real] :
              ( pp(aa_set_real_bool(member_real(A),minus_minus_set_real(S,S5)))
             => ( aa_complex_real(I,aa_real_complex(J,A)) = A ) )
         => ( ! [A: real] :
                ( pp(aa_set_real_bool(member_real(A),minus_minus_set_real(S,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_real_complex(J,A)),minus_811609699411566653omplex(T2,T5))) )
           => ( ! [B: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(T2,T5)))
                 => ( aa_real_complex(J,aa_complex_real(I,B)) = B ) )
             => ( ! [B: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(T2,T5)))
                   => pp(aa_set_real_bool(member_real(aa_complex_real(I,B)),minus_minus_set_real(S,S5))) )
               => ( ! [A: real] :
                      ( pp(aa_set_real_bool(member_real(A),S5))
                     => ( aa_real_real(G3,A) = zero_zero_real ) )
                 => ( ! [B: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B),T5))
                       => ( aa_complex_real(H,B) = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( pp(aa_set_real_bool(member_real(A),S))
                         => ( aa_complex_real(H,aa_real_complex(J,A)) = aa_real_real(G3,A) ) )
                     => ( groups8097168146408367636l_real(G3,S) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8191_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_bool,T5: set_complex,S: set_bool,I: fun_complex_bool,J: fun_bool_complex,T2: set_complex,G3: fun_bool_real,H: fun_complex_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
       => ( ! [A: bool] :
              ( pp(aa_set_bool_bool(member_bool(A),minus_minus_set_bool(S,S5)))
             => ( pp(aa_complex_bool(I,aa_bool_complex(J,A)))
              <=> pp(A) ) )
         => ( ! [A: bool] :
                ( pp(aa_set_bool_bool(member_bool(A),minus_minus_set_bool(S,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_bool_complex(J,A)),minus_811609699411566653omplex(T2,T5))) )
           => ( ! [B: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(T2,T5)))
                 => ( aa_bool_complex(J,aa_complex_bool(I,B)) = B ) )
             => ( ! [B: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(T2,T5)))
                   => pp(aa_set_bool_bool(member_bool(aa_complex_bool(I,B)),minus_minus_set_bool(S,S5))) )
               => ( ! [A: bool] :
                      ( pp(aa_set_bool_bool(member_bool(A),S5))
                     => ( aa_bool_real(G3,A) = zero_zero_real ) )
                 => ( ! [B: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B),T5))
                       => ( aa_complex_real(H,B) = zero_zero_real ) )
                   => ( ! [A: bool] :
                          ( pp(aa_set_bool_bool(member_bool(A),S))
                         => ( aa_complex_real(H,aa_bool_complex(J,A)) = aa_bool_real(G3,A) ) )
                     => ( groups502887708668150281l_real(G3,S) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8192_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_Extended_enat,S: set_real,I: fun_Ex1451620480498224165t_real,J: fun_re6065469302365115155d_enat,T2: set_Extended_enat,G3: fun_real_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T5))
       => ( ! [A: real] :
              ( pp(aa_set_real_bool(member_real(A),minus_minus_set_real(S,S5)))
             => ( aa_Ext2541572767299617110t_real(I,aa_rea1194391308913012860d_enat(J,A)) = A ) )
         => ( ! [A: real] :
                ( pp(aa_set_real_bool(member_real(A),minus_minus_set_real(S,S5)))
               => pp(aa_set3347977610044113549t_bool(member_Extended_enat(aa_rea1194391308913012860d_enat(J,A)),minus_925952699566721837d_enat(T2,T5))) )
           => ( ! [B: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B),minus_925952699566721837d_enat(T2,T5)))
                 => ( aa_rea1194391308913012860d_enat(J,aa_Ext2541572767299617110t_real(I,B)) = B ) )
             => ( ! [B: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B),minus_925952699566721837d_enat(T2,T5)))
                   => pp(aa_set_real_bool(member_real(aa_Ext2541572767299617110t_real(I,B)),minus_minus_set_real(S,S5))) )
               => ( ! [A: real] :
                      ( pp(aa_set_real_bool(member_real(A),S5))
                     => ( aa_real_real(G3,A) = zero_zero_real ) )
                 => ( ! [B: extended_enat] :
                        ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B),T5))
                       => ( aa_Ext2541572767299617110t_real(H,B) = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( pp(aa_set_real_bool(member_real(A),S))
                         => ( aa_Ext2541572767299617110t_real(H,aa_rea1194391308913012860d_enat(J,A)) = aa_real_real(G3,A) ) )
                     => ( groups8097168146408367636l_real(G3,S) = groups4148127829035722712t_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8193_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_bool,T5: set_Extended_enat,S: set_bool,I: fun_Ex8563670519792538868t_bool,J: fun_bo5587879492594653778d_enat,T2: set_Extended_enat,G3: fun_bool_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S5))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T5))
       => ( ! [A: bool] :
              ( pp(aa_set_bool_bool(member_bool(A),minus_minus_set_bool(S,S5)))
             => ( pp(aa_Ext2772979874139974381t_bool(I,aa_boo6911241395695408075d_enat(J,A)))
              <=> pp(A) ) )
         => ( ! [A: bool] :
                ( pp(aa_set_bool_bool(member_bool(A),minus_minus_set_bool(S,S5)))
               => pp(aa_set3347977610044113549t_bool(member_Extended_enat(aa_boo6911241395695408075d_enat(J,A)),minus_925952699566721837d_enat(T2,T5))) )
           => ( ! [B: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B),minus_925952699566721837d_enat(T2,T5)))
                 => ( aa_boo6911241395695408075d_enat(J,aa_Ext2772979874139974381t_bool(I,B)) = B ) )
             => ( ! [B: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B),minus_925952699566721837d_enat(T2,T5)))
                   => pp(aa_set_bool_bool(member_bool(aa_Ext2772979874139974381t_bool(I,B)),minus_minus_set_bool(S,S5))) )
               => ( ! [A: bool] :
                      ( pp(aa_set_bool_bool(member_bool(A),S5))
                     => ( aa_bool_real(G3,A) = zero_zero_real ) )
                 => ( ! [B: extended_enat] :
                        ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B),T5))
                       => ( aa_Ext2541572767299617110t_real(H,B) = zero_zero_real ) )
                   => ( ! [A: bool] :
                          ( pp(aa_set_bool_bool(member_bool(A),S))
                         => ( aa_Ext2541572767299617110t_real(H,aa_boo6911241395695408075d_enat(J,A)) = aa_bool_real(G3,A) ) )
                     => ( groups502887708668150281l_real(G3,S) = groups4148127829035722712t_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8194_sums__mult2__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_na(C),F)),aa_complex_complex(times_times_complex(D),C)))
      <=> pp(aa_complex_bool(sums_complex(F),D)) ) ) ).

% sums_mult2_iff
tff(fact_8195_sums__mult2__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_nb(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,D),C)))
      <=> pp(aa_real_bool(sums_real(F),D)) ) ) ).

% sums_mult2_iff
tff(fact_8196_sums__mult__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hd(C),F)),aa_complex_complex(times_times_complex(C),D)))
      <=> pp(aa_complex_bool(sums_complex(F),D)) ) ) ).

% sums_mult_iff
tff(fact_8197_sums__mult__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_he(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D)))
      <=> pp(aa_real_bool(sums_real(F),D)) ) ) ).

% sums_mult_iff
tff(fact_8198_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F: fun_real_real,B2: real,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))) )
       => ( ( groups8097168146408367636l_real(F,S2) = B2 )
         => ( pp(aa_set_real_bool(member_real(I),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8199_sum__nonneg__leq__bound,axiom,
    ! [S2: set_bool,F: fun_bool_real,B2: real,I: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S2))
     => ( ! [I2: bool] :
            ( pp(aa_set_bool_bool(member_bool(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_bool_real(F,I2))) )
       => ( ( groups502887708668150281l_real(F,S2) = B2 )
         => ( pp(aa_set_bool_bool(member_bool(I),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_bool_real(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8200_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: fun_int_real,B2: real,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))) )
       => ( ( groups8778361861064173332t_real(F,S2) = B2 )
         => ( pp(aa_set_int_bool(member_int(I),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8201_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F: fun_complex_real,B2: real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))) )
       => ( ( groups5808333547571424918x_real(F,S2) = B2 )
         => ( pp(aa_set_complex_bool(member_complex(I),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8202_sum__nonneg__leq__bound,axiom,
    ! [S2: set_Extended_enat,F: fun_Ex1451620480498224165t_real,B2: real,I: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ! [I2: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I2))) )
       => ( ( groups4148127829035722712t_real(F,S2) = B2 )
         => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8203_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F: fun_real_rat,B2: rat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2))) )
       => ( ( groups1300246762558778688al_rat(F,S2) = B2 )
         => ( pp(aa_set_real_bool(member_real(I),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8204_sum__nonneg__leq__bound,axiom,
    ! [S2: set_bool,F: fun_bool_rat,B2: rat,I: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S2))
     => ( ! [I2: bool] :
            ( pp(aa_set_bool_bool(member_bool(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_bool_rat(F,I2))) )
       => ( ( groups5133371213642249141ol_rat(F,S2) = B2 )
         => ( pp(aa_set_bool_bool(member_bool(I),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_bool_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8205_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F: fun_nat_rat,B2: rat,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
       => ( ( groups2906978787729119204at_rat(F,S2) = B2 )
         => ( pp(aa_set_nat_bool(member_nat(I),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8206_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: fun_int_rat,B2: rat,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))) )
       => ( ( groups3906332499630173760nt_rat(F,S2) = B2 )
         => ( pp(aa_set_int_bool(member_int(I),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8207_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F: fun_complex_rat,B2: rat,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2))) )
       => ( ( groups5058264527183730370ex_rat(F,S2) = B2 )
         => ( pp(aa_set_complex_bool(member_complex(I),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8208_sum__nonneg__0,axiom,
    ! [S2: set_real,F: fun_real_real,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))) )
       => ( ( groups8097168146408367636l_real(F,S2) = zero_zero_real )
         => ( pp(aa_set_real_bool(member_real(I),S2))
           => ( aa_real_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8209_sum__nonneg__0,axiom,
    ! [S2: set_bool,F: fun_bool_real,I: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S2))
     => ( ! [I2: bool] :
            ( pp(aa_set_bool_bool(member_bool(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_bool_real(F,I2))) )
       => ( ( groups502887708668150281l_real(F,S2) = zero_zero_real )
         => ( pp(aa_set_bool_bool(member_bool(I),S2))
           => ( aa_bool_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8210_sum__nonneg__0,axiom,
    ! [S2: set_int,F: fun_int_real,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))) )
       => ( ( groups8778361861064173332t_real(F,S2) = zero_zero_real )
         => ( pp(aa_set_int_bool(member_int(I),S2))
           => ( aa_int_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8211_sum__nonneg__0,axiom,
    ! [S2: set_complex,F: fun_complex_real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))) )
       => ( ( groups5808333547571424918x_real(F,S2) = zero_zero_real )
         => ( pp(aa_set_complex_bool(member_complex(I),S2))
           => ( aa_complex_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8212_sum__nonneg__0,axiom,
    ! [S2: set_Extended_enat,F: fun_Ex1451620480498224165t_real,I: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ! [I2: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I2))) )
       => ( ( groups4148127829035722712t_real(F,S2) = zero_zero_real )
         => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),S2))
           => ( aa_Ext2541572767299617110t_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8213_sum__nonneg__0,axiom,
    ! [S2: set_real,F: fun_real_rat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2))) )
       => ( ( groups1300246762558778688al_rat(F,S2) = zero_zero_rat )
         => ( pp(aa_set_real_bool(member_real(I),S2))
           => ( aa_real_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8214_sum__nonneg__0,axiom,
    ! [S2: set_bool,F: fun_bool_rat,I: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S2))
     => ( ! [I2: bool] :
            ( pp(aa_set_bool_bool(member_bool(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_bool_rat(F,I2))) )
       => ( ( groups5133371213642249141ol_rat(F,S2) = zero_zero_rat )
         => ( pp(aa_set_bool_bool(member_bool(I),S2))
           => ( aa_bool_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8215_sum__nonneg__0,axiom,
    ! [S2: set_nat,F: fun_nat_rat,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
       => ( ( groups2906978787729119204at_rat(F,S2) = zero_zero_rat )
         => ( pp(aa_set_nat_bool(member_nat(I),S2))
           => ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8216_sum__nonneg__0,axiom,
    ! [S2: set_int,F: fun_int_rat,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))) )
       => ( ( groups3906332499630173760nt_rat(F,S2) = zero_zero_rat )
         => ( pp(aa_set_int_bool(member_int(I),S2))
           => ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8217_sum__nonneg__0,axiom,
    ! [S2: set_complex,F: fun_complex_rat,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2))) )
       => ( ( groups5058264527183730370ex_rat(F,S2) = zero_zero_rat )
         => ( pp(aa_set_complex_bool(member_complex(I),S2))
           => ( aa_complex_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8218_sum_Ointer__restrict,axiom,
    ! [A3: set_real,G3: fun_real_real,B2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups8097168146408367636l_real(G3,inf_inf_set_real(A3,B2)) = groups8097168146408367636l_real(aa_set8744029839670122869l_real(aTP_Lamm_nc(G3),B2),A3) ) ) ).

% sum.inter_restrict
tff(fact_8219_sum_Ointer__restrict,axiom,
    ! [A3: set_bool,G3: fun_bool_real,B2: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( groups502887708668150281l_real(G3,inf_inf_set_bool(A3,B2)) = groups502887708668150281l_real(aa_set4485503625295242075l_real(aTP_Lamm_nd(G3),B2),A3) ) ) ).

% sum.inter_restrict
tff(fact_8220_sum_Ointer__restrict,axiom,
    ! [A3: set_int,G3: fun_int_real,B2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups8778361861064173332t_real(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)) = groups8778361861064173332t_real(aa_set6366552415685541621t_real(aTP_Lamm_ne(G3),B2),A3) ) ) ).

% sum.inter_restrict
tff(fact_8221_sum_Ointer__restrict,axiom,
    ! [A3: set_complex,G3: fun_complex_real,B2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5808333547571424918x_real(G3,inf_inf_set_complex(A3,B2)) = groups5808333547571424918x_real(aa_set904117813986905849x_real(aTP_Lamm_nf(G3),B2),A3) ) ) ).

% sum.inter_restrict
tff(fact_8222_sum_Ointer__restrict,axiom,
    ! [A3: set_Extended_enat,G3: fun_Ex1451620480498224165t_real,B2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups4148127829035722712t_real(G3,inf_in8357106775501769908d_enat(A3,B2)) = groups4148127829035722712t_real(aa_set3355554880467357083t_real(aTP_Lamm_ng(G3),B2),A3) ) ) ).

% sum.inter_restrict
tff(fact_8223_sum_Ointer__restrict,axiom,
    ! [A3: set_real,G3: fun_real_rat,B2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups1300246762558778688al_rat(G3,inf_inf_set_real(A3,B2)) = groups1300246762558778688al_rat(aa_set4110029769364456865al_rat(aTP_Lamm_nh(G3),B2),A3) ) ) ).

% sum.inter_restrict
tff(fact_8224_sum_Ointer__restrict,axiom,
    ! [A3: set_bool,G3: fun_bool_rat,B2: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( groups5133371213642249141ol_rat(G3,inf_inf_set_bool(A3,B2)) = groups5133371213642249141ol_rat(aa_set1787953457065026183ol_rat(aTP_Lamm_ni(G3),B2),A3) ) ) ).

% sum.inter_restrict
tff(fact_8225_sum_Ointer__restrict,axiom,
    ! [A3: set_int,G3: fun_int_rat,B2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups3906332499630173760nt_rat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)) = groups3906332499630173760nt_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_nj(G3),B2),A3) ) ) ).

% sum.inter_restrict
tff(fact_8226_sum_Ointer__restrict,axiom,
    ! [A3: set_complex,G3: fun_complex_rat,B2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5058264527183730370ex_rat(G3,inf_inf_set_complex(A3,B2)) = groups5058264527183730370ex_rat(aa_set7482584248617139493ex_rat(aTP_Lamm_nk(G3),B2),A3) ) ) ).

% sum.inter_restrict
tff(fact_8227_sum_Ointer__restrict,axiom,
    ! [A3: set_Extended_enat,G3: fun_Ex4152069839811072081at_rat,B2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups1392844769737527556at_rat(G3,inf_in8357106775501769908d_enat(A3,B2)) = groups1392844769737527556at_rat(aa_set8847123550896565447at_rat(aTP_Lamm_nl(G3),B2),A3) ) ) ).

% sum.inter_restrict
tff(fact_8228_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G3: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups8097168146408367636l_real(G3,minus_minus_set_real(A3,collect_real(aTP_Lamm_nm(G3)))) = groups8097168146408367636l_real(G3,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8229_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G3: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups8778361861064173332t_real(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),collect_int(aTP_Lamm_nn(G3)))) = groups8778361861064173332t_real(G3,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8230_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G3: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5808333547571424918x_real(G3,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_no(G3)))) = groups5808333547571424918x_real(G3,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8231_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Extended_enat,G3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups4148127829035722712t_real(G3,minus_925952699566721837d_enat(A3,collec4429806609662206161d_enat(aTP_Lamm_np(G3)))) = groups4148127829035722712t_real(G3,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8232_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G3: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups1300246762558778688al_rat(G3,minus_minus_set_real(A3,collect_real(aTP_Lamm_nq(G3)))) = groups1300246762558778688al_rat(G3,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8233_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G3: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups3906332499630173760nt_rat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),collect_int(aTP_Lamm_nr(G3)))) = groups3906332499630173760nt_rat(G3,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8234_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5058264527183730370ex_rat(G3,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_ns(G3)))) = groups5058264527183730370ex_rat(G3,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8235_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Extended_enat,G3: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups1392844769737527556at_rat(G3,minus_925952699566721837d_enat(A3,collec4429806609662206161d_enat(aTP_Lamm_nt(G3)))) = groups1392844769737527556at_rat(G3,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8236_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G3: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups1935376822645274424al_nat(G3,minus_minus_set_real(A3,collect_real(aTP_Lamm_nu(G3)))) = groups1935376822645274424al_nat(G3,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8237_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G3: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups4541462559716669496nt_nat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),collect_int(aTP_Lamm_nv(G3)))) = groups4541462559716669496nt_nat(G3,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8238_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G3: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G3),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_nw(G3)),set_or1269000886237332187st_nat(M,N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_8239_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G3: fun_nat_real,M: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G3),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hv(G3)),set_or1269000886237332187st_nat(M,N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_8240_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G3: fun_nat_nat,M: nat,K: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G3),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,N),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nx(G3),K)),set_or1269000886237332187st_nat(M,N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_8241_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G3: fun_nat_real,M: nat,K: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G3),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,N),K))) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_hz(G3),K)),set_or1269000886237332187st_nat(M,N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_8242_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( pp(aa_set_real_bool(member_real(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I)))
         => ( ! [I2: real] :
                ( pp(aa_set_real_bool(member_real(I2),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8243_sum__pos2,axiom,
    ! [I5: set_bool,I: bool,F: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,I5))
     => ( pp(aa_set_bool_bool(member_bool(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_bool_real(F,I)))
         => ( ! [I2: bool] :
                ( pp(aa_set_bool_bool(member_bool(I2),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_bool_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups502887708668150281l_real(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8244_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( pp(aa_set_int_bool(member_int(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I)))
         => ( ! [I2: int] :
                ( pp(aa_set_int_bool(member_int(I2),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8245_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( pp(aa_set_complex_bool(member_complex(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I)))
         => ( ! [I2: complex] :
                ( pp(aa_set_complex_bool(member_complex(I2),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8246_sum__pos2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I)))
         => ( ! [I2: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups4148127829035722712t_real(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8247_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( pp(aa_set_real_bool(member_real(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,I)))
         => ( ! [I2: real] :
                ( pp(aa_set_real_bool(member_real(I2),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8248_sum__pos2,axiom,
    ! [I5: set_bool,I: bool,F: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,I5))
     => ( pp(aa_set_bool_bool(member_bool(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_bool_rat(F,I)))
         => ( ! [I2: bool] :
                ( pp(aa_set_bool_bool(member_bool(I2),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_bool_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups5133371213642249141ol_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8249_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( pp(aa_set_nat_bool(member_nat(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I)))
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8250_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( pp(aa_set_int_bool(member_int(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,I)))
         => ( ! [I2: int] :
                ( pp(aa_set_int_bool(member_int(I2),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8251_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( pp(aa_set_complex_bool(member_complex(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I)))
         => ( ! [I2: complex] :
                ( pp(aa_set_complex_bool(member_complex(I2),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8252_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( pp(aa_set_complex_bool(member_complex(I2),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8253_sum__pos,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups4148127829035722712t_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8254_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( pp(aa_set_real_bool(member_real(I2),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8255_sum__pos,axiom,
    ! [I5: set_bool,F: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,I5))
     => ( ( I5 != bot_bot_set_bool )
       => ( ! [I2: bool] :
              ( pp(aa_set_bool_bool(member_bool(I2),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_bool_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups502887708668150281l_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8256_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( pp(aa_set_int_bool(member_int(I2),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8257_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( pp(aa_set_complex_bool(member_complex(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8258_sum__pos,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_Extended_enat_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1392844769737527556at_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8259_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( pp(aa_set_real_bool(member_real(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8260_sum__pos,axiom,
    ! [I5: set_bool,F: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,I5))
     => ( ( I5 != bot_bot_set_bool )
       => ( ! [I2: bool] :
              ( pp(aa_set_bool_bool(member_bool(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_bool_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups5133371213642249141ol_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8261_sum__pos,axiom,
    ! [I5: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( pp(aa_set_nat_bool(member_nat(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8262_sum__bounded__below,axiom,
    ! [A3: set_real,K4: real,F: fun_real_real] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K4),aa_real_real(F,I2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_real(A3))),K4)),groups8097168146408367636l_real(F,A3))) ) ).

% sum_bounded_below
tff(fact_8263_sum__bounded__below,axiom,
    ! [A3: set_bool,K4: real,F: fun_bool_real] :
      ( ! [I2: bool] :
          ( pp(aa_set_bool_bool(member_bool(I2),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K4),aa_bool_real(F,I2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_bool(A3))),K4)),groups502887708668150281l_real(F,A3))) ) ).

% sum_bounded_below
tff(fact_8264_sum__bounded__below,axiom,
    ! [A3: set_complex,K4: real,F: fun_complex_real] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K4),aa_complex_real(F,I2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_complex(A3))),K4)),groups5808333547571424918x_real(F,A3))) ) ).

% sum_bounded_below
tff(fact_8265_sum__bounded__below,axiom,
    ! [A3: set_int,K4: real,F: fun_int_real] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K4),aa_int_real(F,I2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_int(A3))),K4)),groups8778361861064173332t_real(F,A3))) ) ).

% sum_bounded_below
tff(fact_8266_sum__bounded__below,axiom,
    ! [A3: set_real,K4: rat,F: fun_real_rat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K4),aa_real_rat(F,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_real(A3))),K4)),groups1300246762558778688al_rat(F,A3))) ) ).

% sum_bounded_below
tff(fact_8267_sum__bounded__below,axiom,
    ! [A3: set_bool,K4: rat,F: fun_bool_rat] :
      ( ! [I2: bool] :
          ( pp(aa_set_bool_bool(member_bool(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K4),aa_bool_rat(F,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_bool(A3))),K4)),groups5133371213642249141ol_rat(F,A3))) ) ).

% sum_bounded_below
tff(fact_8268_sum__bounded__below,axiom,
    ! [A3: set_complex,K4: rat,F: fun_complex_rat] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K4),aa_complex_rat(F,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_complex(A3))),K4)),groups5058264527183730370ex_rat(F,A3))) ) ).

% sum_bounded_below
tff(fact_8269_sum__bounded__below,axiom,
    ! [A3: set_nat,K4: rat,F: fun_nat_rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K4),aa_nat_rat(F,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A3))),K4)),groups2906978787729119204at_rat(F,A3))) ) ).

% sum_bounded_below
tff(fact_8270_sum__bounded__below,axiom,
    ! [A3: set_int,K4: rat,F: fun_int_rat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K4),aa_int_rat(F,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_int(A3))),K4)),groups3906332499630173760nt_rat(F,A3))) ) ).

% sum_bounded_below
tff(fact_8271_sum__bounded__below,axiom,
    ! [A3: set_real,K4: nat,F: fun_real_nat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),aa_real_nat(F,I2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_real(A3))),K4)),groups1935376822645274424al_nat(F,A3))) ) ).

% sum_bounded_below
tff(fact_8272_sum__bounded__above,axiom,
    ! [A3: set_real,F: fun_real_real,K4: real] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I2)),K4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_real(A3))),K4))) ) ).

% sum_bounded_above
tff(fact_8273_sum__bounded__above,axiom,
    ! [A3: set_bool,F: fun_bool_real,K4: real] :
      ( ! [I2: bool] :
          ( pp(aa_set_bool_bool(member_bool(I2),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_bool_real(F,I2)),K4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups502887708668150281l_real(F,A3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_bool(A3))),K4))) ) ).

% sum_bounded_above
tff(fact_8274_sum__bounded__above,axiom,
    ! [A3: set_complex,F: fun_complex_real,K4: real] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I2)),K4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,A3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_complex(A3))),K4))) ) ).

% sum_bounded_above
tff(fact_8275_sum__bounded__above,axiom,
    ! [A3: set_int,F: fun_int_real,K4: real] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I2)),K4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,A3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_int(A3))),K4))) ) ).

% sum_bounded_above
tff(fact_8276_sum__bounded__above,axiom,
    ! [A3: set_real,F: fun_real_rat,K4: rat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),K4)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_real(A3))),K4))) ) ).

% sum_bounded_above
tff(fact_8277_sum__bounded__above,axiom,
    ! [A3: set_bool,F: fun_bool_rat,K4: rat] :
      ( ! [I2: bool] :
          ( pp(aa_set_bool_bool(member_bool(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_bool_rat(F,I2)),K4)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5133371213642249141ol_rat(F,A3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_bool(A3))),K4))) ) ).

% sum_bounded_above
tff(fact_8278_sum__bounded__above,axiom,
    ! [A3: set_complex,F: fun_complex_rat,K4: rat] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I2)),K4)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_complex(A3))),K4))) ) ).

% sum_bounded_above
tff(fact_8279_sum__bounded__above,axiom,
    ! [A3: set_nat,F: fun_nat_rat,K4: rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),K4)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,A3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A3))),K4))) ) ).

% sum_bounded_above
tff(fact_8280_sum__bounded__above,axiom,
    ! [A3: set_int,F: fun_int_rat,K4: rat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I2)),K4)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_int(A3))),K4))) ) ).

% sum_bounded_above
tff(fact_8281_sum__bounded__above,axiom,
    ! [A3: set_real,F: fun_real_nat,K4: nat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I2)),K4)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,A3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_real(A3))),K4))) ) ).

% sum_bounded_above
tff(fact_8282_sum_Osame__carrier,axiom,
    ! [C2: set_real,A3: set_real,B2: set_real,G3: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),C2))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C2))
         => ( ! [A: real] :
                ( pp(aa_set_real_bool(member_real(A),minus_minus_set_real(C2,A3)))
               => ( aa_real_real(G3,A) = zero_zero_real ) )
           => ( ! [B: real] :
                  ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(C2,B2)))
                 => ( aa_real_real(H,B) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G3,A3) = groups8097168146408367636l_real(H,B2) )
              <=> ( groups8097168146408367636l_real(G3,C2) = groups8097168146408367636l_real(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8283_sum_Osame__carrier,axiom,
    ! [C2: set_bool,A3: set_bool,B2: set_bool,G3: fun_bool_real,H: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,C2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),C2))
       => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(B2),C2))
         => ( ! [A: bool] :
                ( pp(aa_set_bool_bool(member_bool(A),minus_minus_set_bool(C2,A3)))
               => ( aa_bool_real(G3,A) = zero_zero_real ) )
           => ( ! [B: bool] :
                  ( pp(aa_set_bool_bool(member_bool(B),minus_minus_set_bool(C2,B2)))
                 => ( aa_bool_real(H,B) = zero_zero_real ) )
             => ( ( groups502887708668150281l_real(G3,A3) = groups502887708668150281l_real(H,B2) )
              <=> ( groups502887708668150281l_real(G3,C2) = groups502887708668150281l_real(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8284_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A3: set_complex,B2: set_complex,G3: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),C2))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C2))
         => ( ! [A: complex] :
                ( pp(aa_set_complex_bool(member_complex(A),minus_811609699411566653omplex(C2,A3)))
               => ( aa_complex_real(G3,A) = zero_zero_real ) )
           => ( ! [B: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(C2,B2)))
                 => ( aa_complex_real(H,B) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G3,A3) = groups5808333547571424918x_real(H,B2) )
              <=> ( groups5808333547571424918x_real(G3,C2) = groups5808333547571424918x_real(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8285_sum_Osame__carrier,axiom,
    ! [C2: set_Extended_enat,A3: set_Extended_enat,B2: set_Extended_enat,G3: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),C2))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),C2))
         => ( ! [A: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),minus_925952699566721837d_enat(C2,A3)))
               => ( aa_Ext2541572767299617110t_real(G3,A) = zero_zero_real ) )
           => ( ! [B: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B),minus_925952699566721837d_enat(C2,B2)))
                 => ( aa_Ext2541572767299617110t_real(H,B) = zero_zero_real ) )
             => ( ( groups4148127829035722712t_real(G3,A3) = groups4148127829035722712t_real(H,B2) )
              <=> ( groups4148127829035722712t_real(G3,C2) = groups4148127829035722712t_real(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8286_sum_Osame__carrier,axiom,
    ! [C2: set_real,A3: set_real,B2: set_real,G3: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),C2))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C2))
         => ( ! [A: real] :
                ( pp(aa_set_real_bool(member_real(A),minus_minus_set_real(C2,A3)))
               => ( aa_real_rat(G3,A) = zero_zero_rat ) )
           => ( ! [B: real] :
                  ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(C2,B2)))
                 => ( aa_real_rat(H,B) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G3,A3) = groups1300246762558778688al_rat(H,B2) )
              <=> ( groups1300246762558778688al_rat(G3,C2) = groups1300246762558778688al_rat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8287_sum_Osame__carrier,axiom,
    ! [C2: set_bool,A3: set_bool,B2: set_bool,G3: fun_bool_rat,H: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,C2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),C2))
       => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(B2),C2))
         => ( ! [A: bool] :
                ( pp(aa_set_bool_bool(member_bool(A),minus_minus_set_bool(C2,A3)))
               => ( aa_bool_rat(G3,A) = zero_zero_rat ) )
           => ( ! [B: bool] :
                  ( pp(aa_set_bool_bool(member_bool(B),minus_minus_set_bool(C2,B2)))
                 => ( aa_bool_rat(H,B) = zero_zero_rat ) )
             => ( ( groups5133371213642249141ol_rat(G3,A3) = groups5133371213642249141ol_rat(H,B2) )
              <=> ( groups5133371213642249141ol_rat(G3,C2) = groups5133371213642249141ol_rat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8288_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A3: set_complex,B2: set_complex,G3: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),C2))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C2))
         => ( ! [A: complex] :
                ( pp(aa_set_complex_bool(member_complex(A),minus_811609699411566653omplex(C2,A3)))
               => ( aa_complex_rat(G3,A) = zero_zero_rat ) )
           => ( ! [B: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(C2,B2)))
                 => ( aa_complex_rat(H,B) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G3,A3) = groups5058264527183730370ex_rat(H,B2) )
              <=> ( groups5058264527183730370ex_rat(G3,C2) = groups5058264527183730370ex_rat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8289_sum_Osame__carrier,axiom,
    ! [C2: set_Extended_enat,A3: set_Extended_enat,B2: set_Extended_enat,G3: fun_Ex4152069839811072081at_rat,H: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),C2))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),C2))
         => ( ! [A: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),minus_925952699566721837d_enat(C2,A3)))
               => ( aa_Extended_enat_rat(G3,A) = zero_zero_rat ) )
           => ( ! [B: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B),minus_925952699566721837d_enat(C2,B2)))
                 => ( aa_Extended_enat_rat(H,B) = zero_zero_rat ) )
             => ( ( groups1392844769737527556at_rat(G3,A3) = groups1392844769737527556at_rat(H,B2) )
              <=> ( groups1392844769737527556at_rat(G3,C2) = groups1392844769737527556at_rat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8290_sum_Osame__carrier,axiom,
    ! [C2: set_real,A3: set_real,B2: set_real,G3: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,C2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),C2))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C2))
         => ( ! [A: real] :
                ( pp(aa_set_real_bool(member_real(A),minus_minus_set_real(C2,A3)))
               => ( aa_real_nat(G3,A) = zero_zero_nat ) )
           => ( ! [B: real] :
                  ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(C2,B2)))
                 => ( aa_real_nat(H,B) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G3,A3) = groups1935376822645274424al_nat(H,B2) )
              <=> ( groups1935376822645274424al_nat(G3,C2) = groups1935376822645274424al_nat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8291_sum_Osame__carrier,axiom,
    ! [C2: set_bool,A3: set_bool,B2: set_bool,G3: fun_bool_nat,H: fun_bool_nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,C2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),C2))
       => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(B2),C2))
         => ( ! [A: bool] :
                ( pp(aa_set_bool_bool(member_bool(A),minus_minus_set_bool(C2,A3)))
               => ( aa_bool_nat(G3,A) = zero_zero_nat ) )
           => ( ! [B: bool] :
                  ( pp(aa_set_bool_bool(member_bool(B),minus_minus_set_bool(C2,B2)))
                 => ( aa_bool_nat(H,B) = zero_zero_nat ) )
             => ( ( groups5768501273728744877ol_nat(G3,A3) = groups5768501273728744877ol_nat(H,B2) )
              <=> ( groups5768501273728744877ol_nat(G3,C2) = groups5768501273728744877ol_nat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8292_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A3: set_real,B2: set_real,G3: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),C2))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C2))
         => ( ! [A: real] :
                ( pp(aa_set_real_bool(member_real(A),minus_minus_set_real(C2,A3)))
               => ( aa_real_real(G3,A) = zero_zero_real ) )
           => ( ! [B: real] :
                  ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(C2,B2)))
                 => ( aa_real_real(H,B) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G3,C2) = groups8097168146408367636l_real(H,C2) )
               => ( groups8097168146408367636l_real(G3,A3) = groups8097168146408367636l_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8293_sum_Osame__carrierI,axiom,
    ! [C2: set_bool,A3: set_bool,B2: set_bool,G3: fun_bool_real,H: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,C2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),C2))
       => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(B2),C2))
         => ( ! [A: bool] :
                ( pp(aa_set_bool_bool(member_bool(A),minus_minus_set_bool(C2,A3)))
               => ( aa_bool_real(G3,A) = zero_zero_real ) )
           => ( ! [B: bool] :
                  ( pp(aa_set_bool_bool(member_bool(B),minus_minus_set_bool(C2,B2)))
                 => ( aa_bool_real(H,B) = zero_zero_real ) )
             => ( ( groups502887708668150281l_real(G3,C2) = groups502887708668150281l_real(H,C2) )
               => ( groups502887708668150281l_real(G3,A3) = groups502887708668150281l_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8294_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A3: set_complex,B2: set_complex,G3: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),C2))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C2))
         => ( ! [A: complex] :
                ( pp(aa_set_complex_bool(member_complex(A),minus_811609699411566653omplex(C2,A3)))
               => ( aa_complex_real(G3,A) = zero_zero_real ) )
           => ( ! [B: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(C2,B2)))
                 => ( aa_complex_real(H,B) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G3,C2) = groups5808333547571424918x_real(H,C2) )
               => ( groups5808333547571424918x_real(G3,A3) = groups5808333547571424918x_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8295_sum_Osame__carrierI,axiom,
    ! [C2: set_Extended_enat,A3: set_Extended_enat,B2: set_Extended_enat,G3: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),C2))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),C2))
         => ( ! [A: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),minus_925952699566721837d_enat(C2,A3)))
               => ( aa_Ext2541572767299617110t_real(G3,A) = zero_zero_real ) )
           => ( ! [B: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B),minus_925952699566721837d_enat(C2,B2)))
                 => ( aa_Ext2541572767299617110t_real(H,B) = zero_zero_real ) )
             => ( ( groups4148127829035722712t_real(G3,C2) = groups4148127829035722712t_real(H,C2) )
               => ( groups4148127829035722712t_real(G3,A3) = groups4148127829035722712t_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8296_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A3: set_real,B2: set_real,G3: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),C2))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C2))
         => ( ! [A: real] :
                ( pp(aa_set_real_bool(member_real(A),minus_minus_set_real(C2,A3)))
               => ( aa_real_rat(G3,A) = zero_zero_rat ) )
           => ( ! [B: real] :
                  ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(C2,B2)))
                 => ( aa_real_rat(H,B) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G3,C2) = groups1300246762558778688al_rat(H,C2) )
               => ( groups1300246762558778688al_rat(G3,A3) = groups1300246762558778688al_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8297_sum_Osame__carrierI,axiom,
    ! [C2: set_bool,A3: set_bool,B2: set_bool,G3: fun_bool_rat,H: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,C2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),C2))
       => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(B2),C2))
         => ( ! [A: bool] :
                ( pp(aa_set_bool_bool(member_bool(A),minus_minus_set_bool(C2,A3)))
               => ( aa_bool_rat(G3,A) = zero_zero_rat ) )
           => ( ! [B: bool] :
                  ( pp(aa_set_bool_bool(member_bool(B),minus_minus_set_bool(C2,B2)))
                 => ( aa_bool_rat(H,B) = zero_zero_rat ) )
             => ( ( groups5133371213642249141ol_rat(G3,C2) = groups5133371213642249141ol_rat(H,C2) )
               => ( groups5133371213642249141ol_rat(G3,A3) = groups5133371213642249141ol_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8298_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A3: set_complex,B2: set_complex,G3: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),C2))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C2))
         => ( ! [A: complex] :
                ( pp(aa_set_complex_bool(member_complex(A),minus_811609699411566653omplex(C2,A3)))
               => ( aa_complex_rat(G3,A) = zero_zero_rat ) )
           => ( ! [B: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(C2,B2)))
                 => ( aa_complex_rat(H,B) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G3,C2) = groups5058264527183730370ex_rat(H,C2) )
               => ( groups5058264527183730370ex_rat(G3,A3) = groups5058264527183730370ex_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8299_sum_Osame__carrierI,axiom,
    ! [C2: set_Extended_enat,A3: set_Extended_enat,B2: set_Extended_enat,G3: fun_Ex4152069839811072081at_rat,H: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),C2))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),C2))
         => ( ! [A: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),minus_925952699566721837d_enat(C2,A3)))
               => ( aa_Extended_enat_rat(G3,A) = zero_zero_rat ) )
           => ( ! [B: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B),minus_925952699566721837d_enat(C2,B2)))
                 => ( aa_Extended_enat_rat(H,B) = zero_zero_rat ) )
             => ( ( groups1392844769737527556at_rat(G3,C2) = groups1392844769737527556at_rat(H,C2) )
               => ( groups1392844769737527556at_rat(G3,A3) = groups1392844769737527556at_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8300_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A3: set_real,B2: set_real,G3: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,C2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),C2))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C2))
         => ( ! [A: real] :
                ( pp(aa_set_real_bool(member_real(A),minus_minus_set_real(C2,A3)))
               => ( aa_real_nat(G3,A) = zero_zero_nat ) )
           => ( ! [B: real] :
                  ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(C2,B2)))
                 => ( aa_real_nat(H,B) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G3,C2) = groups1935376822645274424al_nat(H,C2) )
               => ( groups1935376822645274424al_nat(G3,A3) = groups1935376822645274424al_nat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8301_sum_Osame__carrierI,axiom,
    ! [C2: set_bool,A3: set_bool,B2: set_bool,G3: fun_bool_nat,H: fun_bool_nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,C2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),C2))
       => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(B2),C2))
         => ( ! [A: bool] :
                ( pp(aa_set_bool_bool(member_bool(A),minus_minus_set_bool(C2,A3)))
               => ( aa_bool_nat(G3,A) = zero_zero_nat ) )
           => ( ! [B: bool] :
                  ( pp(aa_set_bool_bool(member_bool(B),minus_minus_set_bool(C2,B2)))
                 => ( aa_bool_nat(H,B) = zero_zero_nat ) )
             => ( ( groups5768501273728744877ol_nat(G3,C2) = groups5768501273728744877ol_nat(H,C2) )
               => ( groups5768501273728744877ol_nat(G3,A3) = groups5768501273728744877ol_nat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8302_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S: set_complex,G3: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),minus_811609699411566653omplex(T2,S)))
             => ( aa_complex_real(G3,X5) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G3,S) = groups5808333547571424918x_real(G3,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8303_sum_Omono__neutral__left,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S),T2))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),minus_925952699566721837d_enat(T2,S)))
             => ( aa_Ext2541572767299617110t_real(G3,X5) = zero_zero_real ) )
         => ( groups4148127829035722712t_real(G3,S) = groups4148127829035722712t_real(G3,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8304_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S: set_complex,G3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),minus_811609699411566653omplex(T2,S)))
             => ( aa_complex_rat(G3,X5) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G3,S) = groups5058264527183730370ex_rat(G3,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8305_sum_Omono__neutral__left,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S),T2))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),minus_925952699566721837d_enat(T2,S)))
             => ( aa_Extended_enat_rat(G3,X5) = zero_zero_rat ) )
         => ( groups1392844769737527556at_rat(G3,S) = groups1392844769737527556at_rat(G3,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8306_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S: set_complex,G3: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),minus_811609699411566653omplex(T2,S)))
             => ( aa_complex_nat(G3,X5) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G3,S) = groups5693394587270226106ex_nat(G3,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8307_sum_Omono__neutral__left,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S),T2))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),minus_925952699566721837d_enat(T2,S)))
             => ( aa_Extended_enat_nat(G3,X5) = zero_zero_nat ) )
         => ( groups2027974829824023292at_nat(G3,S) = groups2027974829824023292at_nat(G3,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8308_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S: set_complex,G3: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),minus_811609699411566653omplex(T2,S)))
             => ( aa_complex_int(G3,X5) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G3,S) = groups5690904116761175830ex_int(G3,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8309_sum_Omono__neutral__left,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S),T2))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),minus_925952699566721837d_enat(T2,S)))
             => ( aa_Extended_enat_int(G3,X5) = zero_zero_int ) )
         => ( groups2025484359314973016at_int(G3,S) = groups2025484359314973016at_int(G3,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8310_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S: set_nat,G3: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S),T2))
       => ( ! [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S)))
             => ( aa_nat_rat(G3,X5) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G3,S) = groups2906978787729119204at_rat(G3,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8311_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S: set_nat,G3: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S),T2))
       => ( ! [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S)))
             => ( aa_nat_int(G3,X5) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G3,S) = groups3539618377306564664at_int(G3,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8312_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S: set_complex,G3: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),minus_811609699411566653omplex(T2,S)))
             => ( aa_complex_real(G3,X5) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G3,T2) = groups5808333547571424918x_real(G3,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8313_sum_Omono__neutral__right,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S),T2))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),minus_925952699566721837d_enat(T2,S)))
             => ( aa_Ext2541572767299617110t_real(G3,X5) = zero_zero_real ) )
         => ( groups4148127829035722712t_real(G3,T2) = groups4148127829035722712t_real(G3,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8314_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S: set_complex,G3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),minus_811609699411566653omplex(T2,S)))
             => ( aa_complex_rat(G3,X5) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G3,T2) = groups5058264527183730370ex_rat(G3,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8315_sum_Omono__neutral__right,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S),T2))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),minus_925952699566721837d_enat(T2,S)))
             => ( aa_Extended_enat_rat(G3,X5) = zero_zero_rat ) )
         => ( groups1392844769737527556at_rat(G3,T2) = groups1392844769737527556at_rat(G3,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8316_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S: set_complex,G3: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),minus_811609699411566653omplex(T2,S)))
             => ( aa_complex_nat(G3,X5) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G3,T2) = groups5693394587270226106ex_nat(G3,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8317_sum_Omono__neutral__right,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S),T2))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),minus_925952699566721837d_enat(T2,S)))
             => ( aa_Extended_enat_nat(G3,X5) = zero_zero_nat ) )
         => ( groups2027974829824023292at_nat(G3,T2) = groups2027974829824023292at_nat(G3,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8318_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S: set_complex,G3: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),minus_811609699411566653omplex(T2,S)))
             => ( aa_complex_int(G3,X5) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G3,T2) = groups5690904116761175830ex_int(G3,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8319_sum_Omono__neutral__right,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S),T2))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),minus_925952699566721837d_enat(T2,S)))
             => ( aa_Extended_enat_int(G3,X5) = zero_zero_int ) )
         => ( groups2025484359314973016at_int(G3,T2) = groups2025484359314973016at_int(G3,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8320_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S: set_nat,G3: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S),T2))
       => ( ! [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S)))
             => ( aa_nat_rat(G3,X5) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G3,T2) = groups2906978787729119204at_rat(G3,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8321_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S: set_nat,G3: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S),T2))
       => ( ! [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T2),S)))
             => ( aa_nat_int(G3,X5) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G3,T2) = groups3539618377306564664at_int(G3,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8322_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S: set_real,H: fun_real_real,G3: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T2))
       => ( ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),minus_minus_set_real(T2,S)))
             => ( aa_real_real(H,X5) = zero_zero_real ) )
         => ( ! [X5: real] :
                ( pp(aa_set_real_bool(member_real(X5),S))
               => ( aa_real_real(G3,X5) = aa_real_real(H,X5) ) )
           => ( groups8097168146408367636l_real(G3,S) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8323_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_bool,S: set_bool,H: fun_bool_real,G3: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,T2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(S),T2))
       => ( ! [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),minus_minus_set_bool(T2,S)))
             => ( aa_bool_real(H,X5) = zero_zero_real ) )
         => ( ! [X5: bool] :
                ( pp(aa_set_bool_bool(member_bool(X5),S))
               => ( aa_bool_real(G3,X5) = aa_bool_real(H,X5) ) )
           => ( groups502887708668150281l_real(G3,S) = groups502887708668150281l_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8324_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S: set_complex,H: fun_complex_real,G3: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),minus_811609699411566653omplex(T2,S)))
             => ( aa_complex_real(H,X5) = zero_zero_real ) )
         => ( ! [X5: complex] :
                ( pp(aa_set_complex_bool(member_complex(X5),S))
               => ( aa_complex_real(G3,X5) = aa_complex_real(H,X5) ) )
           => ( groups5808333547571424918x_real(G3,S) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8325_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,H: fun_Ex1451620480498224165t_real,G3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S),T2))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),minus_925952699566721837d_enat(T2,S)))
             => ( aa_Ext2541572767299617110t_real(H,X5) = zero_zero_real ) )
         => ( ! [X5: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),S))
               => ( aa_Ext2541572767299617110t_real(G3,X5) = aa_Ext2541572767299617110t_real(H,X5) ) )
           => ( groups4148127829035722712t_real(G3,S) = groups4148127829035722712t_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8326_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S: set_real,H: fun_real_rat,G3: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T2))
       => ( ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),minus_minus_set_real(T2,S)))
             => ( aa_real_rat(H,X5) = zero_zero_rat ) )
         => ( ! [X5: real] :
                ( pp(aa_set_real_bool(member_real(X5),S))
               => ( aa_real_rat(G3,X5) = aa_real_rat(H,X5) ) )
           => ( groups1300246762558778688al_rat(G3,S) = groups1300246762558778688al_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8327_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_bool,S: set_bool,H: fun_bool_rat,G3: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,T2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(S),T2))
       => ( ! [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),minus_minus_set_bool(T2,S)))
             => ( aa_bool_rat(H,X5) = zero_zero_rat ) )
         => ( ! [X5: bool] :
                ( pp(aa_set_bool_bool(member_bool(X5),S))
               => ( aa_bool_rat(G3,X5) = aa_bool_rat(H,X5) ) )
           => ( groups5133371213642249141ol_rat(G3,S) = groups5133371213642249141ol_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8328_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S: set_complex,H: fun_complex_rat,G3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),minus_811609699411566653omplex(T2,S)))
             => ( aa_complex_rat(H,X5) = zero_zero_rat ) )
         => ( ! [X5: complex] :
                ( pp(aa_set_complex_bool(member_complex(X5),S))
               => ( aa_complex_rat(G3,X5) = aa_complex_rat(H,X5) ) )
           => ( groups5058264527183730370ex_rat(G3,S) = groups5058264527183730370ex_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8329_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,H: fun_Ex4152069839811072081at_rat,G3: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S),T2))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),minus_925952699566721837d_enat(T2,S)))
             => ( aa_Extended_enat_rat(H,X5) = zero_zero_rat ) )
         => ( ! [X5: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),S))
               => ( aa_Extended_enat_rat(G3,X5) = aa_Extended_enat_rat(H,X5) ) )
           => ( groups1392844769737527556at_rat(G3,S) = groups1392844769737527556at_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8330_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S: set_real,H: fun_real_nat,G3: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T2))
       => ( ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),minus_minus_set_real(T2,S)))
             => ( aa_real_nat(H,X5) = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( pp(aa_set_real_bool(member_real(X5),S))
               => ( aa_real_nat(G3,X5) = aa_real_nat(H,X5) ) )
           => ( groups1935376822645274424al_nat(G3,S) = groups1935376822645274424al_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8331_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_bool,S: set_bool,H: fun_bool_nat,G3: fun_bool_nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,T2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(S),T2))
       => ( ! [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),minus_minus_set_bool(T2,S)))
             => ( aa_bool_nat(H,X5) = zero_zero_nat ) )
         => ( ! [X5: bool] :
                ( pp(aa_set_bool_bool(member_bool(X5),S))
               => ( aa_bool_nat(G3,X5) = aa_bool_nat(H,X5) ) )
           => ( groups5768501273728744877ol_nat(G3,S) = groups5768501273728744877ol_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8332_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S: set_real,G3: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T2))
       => ( ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),minus_minus_set_real(T2,S)))
             => ( aa_real_real(G3,X5) = zero_zero_real ) )
         => ( ! [X5: real] :
                ( pp(aa_set_real_bool(member_real(X5),S))
               => ( aa_real_real(G3,X5) = aa_real_real(H,X5) ) )
           => ( groups8097168146408367636l_real(G3,T2) = groups8097168146408367636l_real(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8333_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_bool,S: set_bool,G3: fun_bool_real,H: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,T2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(S),T2))
       => ( ! [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),minus_minus_set_bool(T2,S)))
             => ( aa_bool_real(G3,X5) = zero_zero_real ) )
         => ( ! [X5: bool] :
                ( pp(aa_set_bool_bool(member_bool(X5),S))
               => ( aa_bool_real(G3,X5) = aa_bool_real(H,X5) ) )
           => ( groups502887708668150281l_real(G3,T2) = groups502887708668150281l_real(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8334_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S: set_complex,G3: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),minus_811609699411566653omplex(T2,S)))
             => ( aa_complex_real(G3,X5) = zero_zero_real ) )
         => ( ! [X5: complex] :
                ( pp(aa_set_complex_bool(member_complex(X5),S))
               => ( aa_complex_real(G3,X5) = aa_complex_real(H,X5) ) )
           => ( groups5808333547571424918x_real(G3,T2) = groups5808333547571424918x_real(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8335_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S),T2))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),minus_925952699566721837d_enat(T2,S)))
             => ( aa_Ext2541572767299617110t_real(G3,X5) = zero_zero_real ) )
         => ( ! [X5: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),S))
               => ( aa_Ext2541572767299617110t_real(G3,X5) = aa_Ext2541572767299617110t_real(H,X5) ) )
           => ( groups4148127829035722712t_real(G3,T2) = groups4148127829035722712t_real(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8336_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S: set_real,G3: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T2))
       => ( ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),minus_minus_set_real(T2,S)))
             => ( aa_real_rat(G3,X5) = zero_zero_rat ) )
         => ( ! [X5: real] :
                ( pp(aa_set_real_bool(member_real(X5),S))
               => ( aa_real_rat(G3,X5) = aa_real_rat(H,X5) ) )
           => ( groups1300246762558778688al_rat(G3,T2) = groups1300246762558778688al_rat(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8337_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_bool,S: set_bool,G3: fun_bool_rat,H: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,T2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(S),T2))
       => ( ! [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),minus_minus_set_bool(T2,S)))
             => ( aa_bool_rat(G3,X5) = zero_zero_rat ) )
         => ( ! [X5: bool] :
                ( pp(aa_set_bool_bool(member_bool(X5),S))
               => ( aa_bool_rat(G3,X5) = aa_bool_rat(H,X5) ) )
           => ( groups5133371213642249141ol_rat(G3,T2) = groups5133371213642249141ol_rat(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8338_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S: set_complex,G3: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),minus_811609699411566653omplex(T2,S)))
             => ( aa_complex_rat(G3,X5) = zero_zero_rat ) )
         => ( ! [X5: complex] :
                ( pp(aa_set_complex_bool(member_complex(X5),S))
               => ( aa_complex_rat(G3,X5) = aa_complex_rat(H,X5) ) )
           => ( groups5058264527183730370ex_rat(G3,T2) = groups5058264527183730370ex_rat(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8339_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: fun_Ex4152069839811072081at_rat,H: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S),T2))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),minus_925952699566721837d_enat(T2,S)))
             => ( aa_Extended_enat_rat(G3,X5) = zero_zero_rat ) )
         => ( ! [X5: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),S))
               => ( aa_Extended_enat_rat(G3,X5) = aa_Extended_enat_rat(H,X5) ) )
           => ( groups1392844769737527556at_rat(G3,T2) = groups1392844769737527556at_rat(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8340_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S: set_real,G3: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T2))
       => ( ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),minus_minus_set_real(T2,S)))
             => ( aa_real_nat(G3,X5) = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( pp(aa_set_real_bool(member_real(X5),S))
               => ( aa_real_nat(G3,X5) = aa_real_nat(H,X5) ) )
           => ( groups1935376822645274424al_nat(G3,T2) = groups1935376822645274424al_nat(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8341_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_bool,S: set_bool,G3: fun_bool_nat,H: fun_bool_nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,T2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(S),T2))
       => ( ! [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),minus_minus_set_bool(T2,S)))
             => ( aa_bool_nat(G3,X5) = zero_zero_nat ) )
         => ( ! [X5: bool] :
                ( pp(aa_set_bool_bool(member_bool(X5),S))
               => ( aa_bool_nat(G3,X5) = aa_bool_nat(H,X5) ) )
           => ( groups5768501273728744877ol_nat(G3,T2) = groups5768501273728744877ol_nat(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8342_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A3: set_complex,G3: fun_complex_real] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
       => ( groups5808333547571424918x_real(G3,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G3,minus_811609699411566653omplex(A3,B2))),groups5808333547571424918x_real(G3,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8343_sum_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,G3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
       => ( groups4148127829035722712t_real(G3,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups4148127829035722712t_real(G3,minus_925952699566721837d_enat(A3,B2))),groups4148127829035722712t_real(G3,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8344_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A3: set_complex,G3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
       => ( groups5058264527183730370ex_rat(G3,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(G3,minus_811609699411566653omplex(A3,B2))),groups5058264527183730370ex_rat(G3,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8345_sum_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,G3: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
       => ( groups1392844769737527556at_rat(G3,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups1392844769737527556at_rat(G3,minus_925952699566721837d_enat(A3,B2))),groups1392844769737527556at_rat(G3,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8346_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A3: set_complex,G3: fun_complex_nat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
       => ( groups5693394587270226106ex_nat(G3,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G3,minus_811609699411566653omplex(A3,B2))),groups5693394587270226106ex_nat(G3,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8347_sum_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,G3: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
       => ( groups2027974829824023292at_nat(G3,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2027974829824023292at_nat(G3,minus_925952699566721837d_enat(A3,B2))),groups2027974829824023292at_nat(G3,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8348_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A3: set_complex,G3: fun_complex_int] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
       => ( groups5690904116761175830ex_int(G3,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G3,minus_811609699411566653omplex(A3,B2))),groups5690904116761175830ex_int(G3,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8349_sum_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,G3: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
       => ( groups2025484359314973016at_int(G3,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups2025484359314973016at_int(G3,minus_925952699566721837d_enat(A3,B2))),groups2025484359314973016at_int(G3,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8350_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A3: set_nat,G3: fun_nat_rat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A3))
       => ( groups2906978787729119204at_rat(G3,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G3,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2))),groups2906978787729119204at_rat(G3,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8351_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A3: set_nat,G3: fun_nat_int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A3))
       => ( groups3539618377306564664at_int(G3,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G3,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2))),groups3539618377306564664at_int(G3,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8352_sum__diff,axiom,
    ! [A3: set_complex,B2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A3))
       => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A3,B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A3)),groups5808333547571424918x_real(F,B2)) ) ) ) ).

% sum_diff
tff(fact_8353_sum__diff,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),A3))
       => ( groups4148127829035722712t_real(F,minus_925952699566721837d_enat(A3,B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups4148127829035722712t_real(F,A3)),groups4148127829035722712t_real(F,B2)) ) ) ) ).

% sum_diff
tff(fact_8354_sum__diff,axiom,
    ! [A3: set_complex,B2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A3))
       => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A3,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_8355_sum__diff,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),A3))
       => ( groups1392844769737527556at_rat(F,minus_925952699566721837d_enat(A3,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups1392844769737527556at_rat(F,A3)),groups1392844769737527556at_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_8356_sum__diff,axiom,
    ! [A3: set_complex,B2: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A3))
       => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A3,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A3)),groups5690904116761175830ex_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_8357_sum__diff,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),A3))
       => ( groups2025484359314973016at_int(F,minus_925952699566721837d_enat(A3,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups2025484359314973016at_int(F,A3)),groups2025484359314973016at_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_8358_sum__diff,axiom,
    ! [A3: set_nat,B2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A3))
       => ( groups2906978787729119204at_rat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups2906978787729119204at_rat(F,A3)),groups2906978787729119204at_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_8359_sum__diff,axiom,
    ! [A3: set_nat,B2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A3))
       => ( groups3539618377306564664at_int(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,A3)),groups3539618377306564664at_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_8360_sum__diff,axiom,
    ! [A3: set_int,B2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A3))
       => ( groups8778361861064173332t_real(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A3)),groups8778361861064173332t_real(F,B2)) ) ) ) ).

% sum_diff
tff(fact_8361_sum__diff,axiom,
    ! [A3: set_int,B2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A3))
       => ( groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_8362_sum_Omono__neutral__cong,axiom,
    ! [T2: set_real,S: set_real,H: fun_real_real,G3: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(finite_finite_real,S))
       => ( ! [I2: real] :
              ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(T2,S)))
             => ( aa_real_real(H,I2) = zero_zero_real ) )
         => ( ! [I2: real] :
                ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(S,T2)))
               => ( aa_real_real(G3,I2) = zero_zero_real ) )
           => ( ! [X5: real] :
                  ( pp(aa_set_real_bool(member_real(X5),inf_inf_set_real(S,T2)))
                 => ( aa_real_real(G3,X5) = aa_real_real(H,X5) ) )
             => ( groups8097168146408367636l_real(G3,S) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8363_sum_Omono__neutral__cong,axiom,
    ! [T2: set_bool,S: set_bool,H: fun_bool_real,G3: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,T2))
     => ( pp(aa_set_bool_bool(finite_finite_bool,S))
       => ( ! [I2: bool] :
              ( pp(aa_set_bool_bool(member_bool(I2),minus_minus_set_bool(T2,S)))
             => ( aa_bool_real(H,I2) = zero_zero_real ) )
         => ( ! [I2: bool] :
                ( pp(aa_set_bool_bool(member_bool(I2),minus_minus_set_bool(S,T2)))
               => ( aa_bool_real(G3,I2) = zero_zero_real ) )
           => ( ! [X5: bool] :
                  ( pp(aa_set_bool_bool(member_bool(X5),inf_inf_set_bool(S,T2)))
                 => ( aa_bool_real(G3,X5) = aa_bool_real(H,X5) ) )
             => ( groups502887708668150281l_real(G3,S) = groups502887708668150281l_real(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8364_sum_Omono__neutral__cong,axiom,
    ! [T2: set_int,S: set_int,H: fun_int_real,G3: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(finite_finite_int,S))
       => ( ! [I2: int] :
              ( pp(aa_set_int_bool(member_int(I2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T2),S)))
             => ( aa_int_real(H,I2) = zero_zero_real ) )
         => ( ! [I2: int] :
                ( pp(aa_set_int_bool(member_int(I2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),T2)))
               => ( aa_int_real(G3,I2) = zero_zero_real ) )
           => ( ! [X5: int] :
                  ( pp(aa_set_int_bool(member_int(X5),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T2)))
                 => ( aa_int_real(G3,X5) = aa_int_real(H,X5) ) )
             => ( groups8778361861064173332t_real(G3,S) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8365_sum_Omono__neutral__cong,axiom,
    ! [T2: set_complex,S: set_complex,H: fun_complex_real,G3: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
       => ( ! [I2: complex] :
              ( pp(aa_set_complex_bool(member_complex(I2),minus_811609699411566653omplex(T2,S)))
             => ( aa_complex_real(H,I2) = zero_zero_real ) )
         => ( ! [I2: complex] :
                ( pp(aa_set_complex_bool(member_complex(I2),minus_811609699411566653omplex(S,T2)))
               => ( aa_complex_real(G3,I2) = zero_zero_real ) )
           => ( ! [X5: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X5),inf_inf_set_complex(S,T2)))
                 => ( aa_complex_real(G3,X5) = aa_complex_real(H,X5) ) )
             => ( groups5808333547571424918x_real(G3,S) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8366_sum_Omono__neutral__cong,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,H: fun_Ex1451620480498224165t_real,G3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
       => ( ! [I2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),minus_925952699566721837d_enat(T2,S)))
             => ( aa_Ext2541572767299617110t_real(H,I2) = zero_zero_real ) )
         => ( ! [I2: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),minus_925952699566721837d_enat(S,T2)))
               => ( aa_Ext2541572767299617110t_real(G3,I2) = zero_zero_real ) )
           => ( ! [X5: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),inf_in8357106775501769908d_enat(S,T2)))
                 => ( aa_Ext2541572767299617110t_real(G3,X5) = aa_Ext2541572767299617110t_real(H,X5) ) )
             => ( groups4148127829035722712t_real(G3,S) = groups4148127829035722712t_real(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8367_sum_Omono__neutral__cong,axiom,
    ! [T2: set_real,S: set_real,H: fun_real_rat,G3: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(finite_finite_real,S))
       => ( ! [I2: real] :
              ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(T2,S)))
             => ( aa_real_rat(H,I2) = zero_zero_rat ) )
         => ( ! [I2: real] :
                ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(S,T2)))
               => ( aa_real_rat(G3,I2) = zero_zero_rat ) )
           => ( ! [X5: real] :
                  ( pp(aa_set_real_bool(member_real(X5),inf_inf_set_real(S,T2)))
                 => ( aa_real_rat(G3,X5) = aa_real_rat(H,X5) ) )
             => ( groups1300246762558778688al_rat(G3,S) = groups1300246762558778688al_rat(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8368_sum_Omono__neutral__cong,axiom,
    ! [T2: set_bool,S: set_bool,H: fun_bool_rat,G3: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,T2))
     => ( pp(aa_set_bool_bool(finite_finite_bool,S))
       => ( ! [I2: bool] :
              ( pp(aa_set_bool_bool(member_bool(I2),minus_minus_set_bool(T2,S)))
             => ( aa_bool_rat(H,I2) = zero_zero_rat ) )
         => ( ! [I2: bool] :
                ( pp(aa_set_bool_bool(member_bool(I2),minus_minus_set_bool(S,T2)))
               => ( aa_bool_rat(G3,I2) = zero_zero_rat ) )
           => ( ! [X5: bool] :
                  ( pp(aa_set_bool_bool(member_bool(X5),inf_inf_set_bool(S,T2)))
                 => ( aa_bool_rat(G3,X5) = aa_bool_rat(H,X5) ) )
             => ( groups5133371213642249141ol_rat(G3,S) = groups5133371213642249141ol_rat(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8369_sum_Omono__neutral__cong,axiom,
    ! [T2: set_int,S: set_int,H: fun_int_rat,G3: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(finite_finite_int,S))
       => ( ! [I2: int] :
              ( pp(aa_set_int_bool(member_int(I2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T2),S)))
             => ( aa_int_rat(H,I2) = zero_zero_rat ) )
         => ( ! [I2: int] :
                ( pp(aa_set_int_bool(member_int(I2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),T2)))
               => ( aa_int_rat(G3,I2) = zero_zero_rat ) )
           => ( ! [X5: int] :
                  ( pp(aa_set_int_bool(member_int(X5),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,S),T2)))
                 => ( aa_int_rat(G3,X5) = aa_int_rat(H,X5) ) )
             => ( groups3906332499630173760nt_rat(G3,S) = groups3906332499630173760nt_rat(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8370_sum_Omono__neutral__cong,axiom,
    ! [T2: set_complex,S: set_complex,H: fun_complex_rat,G3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
       => ( ! [I2: complex] :
              ( pp(aa_set_complex_bool(member_complex(I2),minus_811609699411566653omplex(T2,S)))
             => ( aa_complex_rat(H,I2) = zero_zero_rat ) )
         => ( ! [I2: complex] :
                ( pp(aa_set_complex_bool(member_complex(I2),minus_811609699411566653omplex(S,T2)))
               => ( aa_complex_rat(G3,I2) = zero_zero_rat ) )
           => ( ! [X5: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X5),inf_inf_set_complex(S,T2)))
                 => ( aa_complex_rat(G3,X5) = aa_complex_rat(H,X5) ) )
             => ( groups5058264527183730370ex_rat(G3,S) = groups5058264527183730370ex_rat(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8371_sum_Omono__neutral__cong,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,H: fun_Ex4152069839811072081at_rat,G3: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
       => ( ! [I2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),minus_925952699566721837d_enat(T2,S)))
             => ( aa_Extended_enat_rat(H,I2) = zero_zero_rat ) )
         => ( ! [I2: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),minus_925952699566721837d_enat(S,T2)))
               => ( aa_Extended_enat_rat(G3,I2) = zero_zero_rat ) )
           => ( ! [X5: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),inf_in8357106775501769908d_enat(S,T2)))
                 => ( aa_Extended_enat_rat(G3,X5) = aa_Extended_enat_rat(H,X5) ) )
             => ( groups1392844769737527556at_rat(G3,S) = groups1392844769737527556at_rat(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8372_sum_Ounion__inter,axiom,
    ! [A3: set_int,B2: set_int,G3: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2))),groups8778361861064173332t_real(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G3,A3)),groups8778361861064173332t_real(G3,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8373_sum_Ounion__inter,axiom,
    ! [A3: set_complex,B2: set_complex,G3: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G3,sup_sup_set_complex(A3,B2))),groups5808333547571424918x_real(G3,inf_inf_set_complex(A3,B2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G3,A3)),groups5808333547571424918x_real(G3,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8374_sum_Ounion__inter,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups4148127829035722712t_real(G3,sup_su4489774667511045786d_enat(A3,B2))),groups4148127829035722712t_real(G3,inf_in8357106775501769908d_enat(A3,B2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups4148127829035722712t_real(G3,A3)),groups4148127829035722712t_real(G3,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8375_sum_Ounion__inter,axiom,
    ! [A3: set_int,B2: set_int,G3: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2))),groups3906332499630173760nt_rat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G3,A3)),groups3906332499630173760nt_rat(G3,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8376_sum_Ounion__inter,axiom,
    ! [A3: set_complex,B2: set_complex,G3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(G3,sup_sup_set_complex(A3,B2))),groups5058264527183730370ex_rat(G3,inf_inf_set_complex(A3,B2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(G3,A3)),groups5058264527183730370ex_rat(G3,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8377_sum_Ounion__inter,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups1392844769737527556at_rat(G3,sup_su4489774667511045786d_enat(A3,B2))),groups1392844769737527556at_rat(G3,inf_in8357106775501769908d_enat(A3,B2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups1392844769737527556at_rat(G3,A3)),groups1392844769737527556at_rat(G3,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8378_sum_Ounion__inter,axiom,
    ! [A3: set_int,B2: set_int,G3: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2))),groups4541462559716669496nt_nat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G3,A3)),groups4541462559716669496nt_nat(G3,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8379_sum_Ounion__inter,axiom,
    ! [A3: set_complex,B2: set_complex,G3: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G3,sup_sup_set_complex(A3,B2))),groups5693394587270226106ex_nat(G3,inf_inf_set_complex(A3,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G3,A3)),groups5693394587270226106ex_nat(G3,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8380_sum_Ounion__inter,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2027974829824023292at_nat(G3,sup_su4489774667511045786d_enat(A3,B2))),groups2027974829824023292at_nat(G3,inf_in8357106775501769908d_enat(A3,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2027974829824023292at_nat(G3,A3)),groups2027974829824023292at_nat(G3,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8381_sum_Ounion__inter,axiom,
    ! [A3: set_complex,B2: set_complex,G3: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G3,sup_sup_set_complex(A3,B2))),groups5690904116761175830ex_int(G3,inf_inf_set_complex(A3,B2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G3,A3)),groups5690904116761175830ex_int(G3,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8382_sum_OInt__Diff,axiom,
    ! [A3: set_int,G3: fun_int_real,B2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups8778361861064173332t_real(G3,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))),groups8778361861064173332t_real(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2))) ) ) ).

% sum.Int_Diff
tff(fact_8383_sum_OInt__Diff,axiom,
    ! [A3: set_complex,G3: fun_complex_real,B2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5808333547571424918x_real(G3,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G3,inf_inf_set_complex(A3,B2))),groups5808333547571424918x_real(G3,minus_811609699411566653omplex(A3,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8384_sum_OInt__Diff,axiom,
    ! [A3: set_Extended_enat,G3: fun_Ex1451620480498224165t_real,B2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups4148127829035722712t_real(G3,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups4148127829035722712t_real(G3,inf_in8357106775501769908d_enat(A3,B2))),groups4148127829035722712t_real(G3,minus_925952699566721837d_enat(A3,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8385_sum_OInt__Diff,axiom,
    ! [A3: set_int,G3: fun_int_rat,B2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups3906332499630173760nt_rat(G3,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))),groups3906332499630173760nt_rat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2))) ) ) ).

% sum.Int_Diff
tff(fact_8386_sum_OInt__Diff,axiom,
    ! [A3: set_complex,G3: fun_complex_rat,B2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5058264527183730370ex_rat(G3,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(G3,inf_inf_set_complex(A3,B2))),groups5058264527183730370ex_rat(G3,minus_811609699411566653omplex(A3,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8387_sum_OInt__Diff,axiom,
    ! [A3: set_Extended_enat,G3: fun_Ex4152069839811072081at_rat,B2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups1392844769737527556at_rat(G3,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups1392844769737527556at_rat(G3,inf_in8357106775501769908d_enat(A3,B2))),groups1392844769737527556at_rat(G3,minus_925952699566721837d_enat(A3,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8388_sum_OInt__Diff,axiom,
    ! [A3: set_int,G3: fun_int_nat,B2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups4541462559716669496nt_nat(G3,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))),groups4541462559716669496nt_nat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2))) ) ) ).

% sum.Int_Diff
tff(fact_8389_sum_OInt__Diff,axiom,
    ! [A3: set_complex,G3: fun_complex_nat,B2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5693394587270226106ex_nat(G3,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G3,inf_inf_set_complex(A3,B2))),groups5693394587270226106ex_nat(G3,minus_811609699411566653omplex(A3,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8390_sum_OInt__Diff,axiom,
    ! [A3: set_Extended_enat,G3: fun_Ex7660298398204468809at_nat,B2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups2027974829824023292at_nat(G3,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2027974829824023292at_nat(G3,inf_in8357106775501769908d_enat(A3,B2))),groups2027974829824023292at_nat(G3,minus_925952699566721837d_enat(A3,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8391_sum_OInt__Diff,axiom,
    ! [A3: set_complex,G3: fun_complex_int,B2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5690904116761175830ex_int(G3,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G3,inf_inf_set_complex(A3,B2))),groups5690904116761175830ex_int(G3,minus_811609699411566653omplex(A3,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8392_sums__mult__D,axiom,
    ! [C: real,F: fun_nat_real,A2: real] :
      ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_he(C),F)),A2))
     => ( ( C != zero_zero_real )
       => pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C))) ) ) ).

% sums_mult_D
tff(fact_8393_sums__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex,A2: complex] :
      ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hd(C),F)),A2))
     => ( ( C != zero_zero_complex )
       => pp(aa_complex_bool(sums_complex(F),aa_complex_complex(divide1717551699836669952omplex(A2),C))) ) ) ).

% sums_mult_D
tff(fact_8394_sums__Suc__imp,axiom,
    ! [F: fun_nat_real,S2: real] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aTP_Lamm_hv(F)),S2))
       => pp(aa_real_bool(sums_real(F),S2)) ) ) ).

% sums_Suc_imp
tff(fact_8395_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_hv(F)),L))
     => pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,L),aa_nat_real(F,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_8396_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( pp(aa_nat_bool(sums_nat(aTP_Lamm_nw(F)),L))
     => pp(aa_nat_bool(sums_nat(F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),aa_nat_nat(F,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_8397_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( pp(aa_int_bool(sums_int(aTP_Lamm_ny(F)),L))
     => pp(aa_int_bool(sums_int(F),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),aa_nat_int(F,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_8398_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S2: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_hv(F)),S2))
    <=> pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S2),aa_nat_real(F,zero_zero_nat)))) ) ).

% sums_Suc_iff
tff(fact_8399_sums__zero__iff__shift,axiom,
    ! [N: nat,F: fun_nat_real,S2: real] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => ( aa_nat_real(F,I2) = zero_zero_real ) )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_nz(N),F)),S2))
      <=> pp(aa_real_bool(sums_real(F),S2)) ) ) ).

% sums_zero_iff_shift
tff(fact_8400_sum_OIf__cases,axiom,
    ! [A3: set_real,P: fun_real_bool,H: fun_real_real,G3: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_oa(P),H),G3),A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8097168146408367636l_real(H,inf_inf_set_real(A3,collect_real(P)))),groups8097168146408367636l_real(G3,inf_inf_set_real(A3,uminus612125837232591019t_real(collect_real(P))))) ) ) ).

% sum.If_cases
tff(fact_8401_sum_OIf__cases,axiom,
    ! [A3: set_int,P: fun_int_bool,H: fun_int_real,G3: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_ob(P),H),G3),A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(H,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(P)))),groups8778361861064173332t_real(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),uminus1532241313380277803et_int(collect_int(P))))) ) ) ).

% sum.If_cases
tff(fact_8402_sum_OIf__cases,axiom,
    ! [A3: set_complex,P: fun_complex_bool,H: fun_complex_real,G3: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_oc(P),H),G3),A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(H,inf_inf_set_complex(A3,collect_complex(P)))),groups5808333547571424918x_real(G3,inf_inf_set_complex(A3,uminus8566677241136511917omplex(collect_complex(P))))) ) ) ).

% sum.If_cases
tff(fact_8403_sum_OIf__cases,axiom,
    ! [A3: set_Extended_enat,P: fun_Ex8563670519792538868t_bool,H: fun_Ex1451620480498224165t_real,G3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aa_fun967516747568145260t_real(aTP_Lamm_od(P),H),G3),A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups4148127829035722712t_real(H,inf_in8357106775501769908d_enat(A3,collec4429806609662206161d_enat(P)))),groups4148127829035722712t_real(G3,inf_in8357106775501769908d_enat(A3,uminus417252749190364093d_enat(collec4429806609662206161d_enat(P))))) ) ) ).

% sum.If_cases
tff(fact_8404_sum_OIf__cases,axiom,
    ! [A3: set_real,P: fun_real_bool,H: fun_real_rat,G3: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_oe(P),H),G3),A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups1300246762558778688al_rat(H,inf_inf_set_real(A3,collect_real(P)))),groups1300246762558778688al_rat(G3,inf_inf_set_real(A3,uminus612125837232591019t_real(collect_real(P))))) ) ) ).

% sum.If_cases
tff(fact_8405_sum_OIf__cases,axiom,
    ! [A3: set_int,P: fun_int_bool,H: fun_int_rat,G3: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_of(P),H),G3),A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(H,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(P)))),groups3906332499630173760nt_rat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),uminus1532241313380277803et_int(collect_int(P))))) ) ) ).

% sum.If_cases
tff(fact_8406_sum_OIf__cases,axiom,
    ! [A3: set_complex,P: fun_complex_bool,H: fun_complex_rat,G3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_og(P),H),G3),A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(H,inf_inf_set_complex(A3,collect_complex(P)))),groups5058264527183730370ex_rat(G3,inf_inf_set_complex(A3,uminus8566677241136511917omplex(collect_complex(P))))) ) ) ).

% sum.If_cases
tff(fact_8407_sum_OIf__cases,axiom,
    ! [A3: set_Extended_enat,P: fun_Ex8563670519792538868t_bool,H: fun_Ex4152069839811072081at_rat,G3: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups1392844769737527556at_rat(aa_fun9057213682095532284at_rat(aa_fun4607285249076117784at_rat(aTP_Lamm_oh(P),H),G3),A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups1392844769737527556at_rat(H,inf_in8357106775501769908d_enat(A3,collec4429806609662206161d_enat(P)))),groups1392844769737527556at_rat(G3,inf_in8357106775501769908d_enat(A3,uminus417252749190364093d_enat(collec4429806609662206161d_enat(P))))) ) ) ).

% sum.If_cases
tff(fact_8408_sum_OIf__cases,axiom,
    ! [A3: set_real,P: fun_real_bool,H: fun_real_nat,G3: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_oi(P),H),G3),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups1935376822645274424al_nat(H,inf_inf_set_real(A3,collect_real(P)))),groups1935376822645274424al_nat(G3,inf_inf_set_real(A3,uminus612125837232591019t_real(collect_real(P))))) ) ) ).

% sum.If_cases
tff(fact_8409_sum_OIf__cases,axiom,
    ! [A3: set_int,P: fun_int_bool,H: fun_int_nat,G3: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_oj(P),H),G3),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(H,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),collect_int(P)))),groups4541462559716669496nt_nat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),uminus1532241313380277803et_int(collect_int(P))))) ) ) ).

% sum.If_cases
tff(fact_8410_sum__power__add,axiom,
    ! [X: complex,M: nat,I5: set_nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_ok(X),M)),I5) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),I5)) ).

% sum_power_add
tff(fact_8411_sum__power__add,axiom,
    ! [X: rat,M: nat,I5: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_ol(X),M),I5) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),M)),groups2906978787729119204at_rat(power_power_rat(X),I5)) ).

% sum_power_add
tff(fact_8412_sum__power__add,axiom,
    ! [X: int,M: nat,I5: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_om(X),M),I5) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),M)),groups3539618377306564664at_int(power_power_int(X),I5)) ).

% sum_power_add
tff(fact_8413_sum__power__add,axiom,
    ! [X: real,M: nat,I5: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_on(X),M)),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),I5)) ).

% sum_power_add
tff(fact_8414_sum__mono2,axiom,
    ! [B2: set_real,A3: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2))
       => ( ! [B: real] :
              ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A3)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,B))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A3)),groups8097168146408367636l_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8415_sum__mono2,axiom,
    ! [B2: set_bool,A3: set_bool,F: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,B2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),B2))
       => ( ! [B: bool] :
              ( pp(aa_set_bool_bool(member_bool(B),minus_minus_set_bool(B2,A3)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_bool_real(F,B))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups502887708668150281l_real(F,A3)),groups502887708668150281l_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8416_sum__mono2,axiom,
    ! [B2: set_complex,A3: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),B2))
       => ( ! [B: complex] :
              ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B2,A3)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,B))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,A3)),groups5808333547571424918x_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8417_sum__mono2,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),B2))
       => ( ! [B: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B),minus_925952699566721837d_enat(B2,A3)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,B))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups4148127829035722712t_real(F,A3)),groups4148127829035722712t_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8418_sum__mono2,axiom,
    ! [B2: set_real,A3: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2))
       => ( ! [B: real] :
              ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A3)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,B))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A3)),groups1300246762558778688al_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8419_sum__mono2,axiom,
    ! [B2: set_bool,A3: set_bool,F: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,B2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),B2))
       => ( ! [B: bool] :
              ( pp(aa_set_bool_bool(member_bool(B),minus_minus_set_bool(B2,A3)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_bool_rat(F,B))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5133371213642249141ol_rat(F,A3)),groups5133371213642249141ol_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8420_sum__mono2,axiom,
    ! [B2: set_complex,A3: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),B2))
       => ( ! [B: complex] :
              ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B2,A3)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8421_sum__mono2,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),B2))
       => ( ! [B: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B),minus_925952699566721837d_enat(B2,A3)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,B))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1392844769737527556at_rat(F,A3)),groups1392844769737527556at_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8422_sum__mono2,axiom,
    ! [B2: set_nat,A3: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
       => ( ! [B: nat] :
              ( pp(aa_set_nat_bool(member_nat(B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A3)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,B))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,A3)),groups2906978787729119204at_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8423_sum__mono2,axiom,
    ! [B2: set_real,A3: set_real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2))
       => ( ! [B: real] :
              ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A3)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,B))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,A3)),groups1935376822645274424al_nat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8424_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B2: set_int,G3: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)))
             => ( aa_int_real(G3,X5) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G3,A3)),groups8778361861064173332t_real(G3,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8425_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_complex,B2: set_complex,G3: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),inf_inf_set_complex(A3,B2)))
             => ( aa_complex_real(G3,X5) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G3,sup_sup_set_complex(A3,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G3,A3)),groups5808333547571424918x_real(G3,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8426_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),inf_in8357106775501769908d_enat(A3,B2)))
             => ( aa_Ext2541572767299617110t_real(G3,X5) = zero_zero_real ) )
         => ( groups4148127829035722712t_real(G3,sup_su4489774667511045786d_enat(A3,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups4148127829035722712t_real(G3,A3)),groups4148127829035722712t_real(G3,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8427_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B2: set_int,G3: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)))
             => ( aa_int_rat(G3,X5) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G3,A3)),groups3906332499630173760nt_rat(G3,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8428_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_complex,B2: set_complex,G3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),inf_inf_set_complex(A3,B2)))
             => ( aa_complex_rat(G3,X5) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G3,sup_sup_set_complex(A3,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(G3,A3)),groups5058264527183730370ex_rat(G3,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8429_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),inf_in8357106775501769908d_enat(A3,B2)))
             => ( aa_Extended_enat_rat(G3,X5) = zero_zero_rat ) )
         => ( groups1392844769737527556at_rat(G3,sup_su4489774667511045786d_enat(A3,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups1392844769737527556at_rat(G3,A3)),groups1392844769737527556at_rat(G3,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8430_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B2: set_int,G3: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2)))
             => ( aa_int_nat(G3,X5) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G3,A3)),groups4541462559716669496nt_nat(G3,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8431_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_complex,B2: set_complex,G3: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),inf_inf_set_complex(A3,B2)))
             => ( aa_complex_nat(G3,X5) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G3,sup_sup_set_complex(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G3,A3)),groups5693394587270226106ex_nat(G3,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8432_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),inf_in8357106775501769908d_enat(A3,B2)))
             => ( aa_Extended_enat_nat(G3,X5) = zero_zero_nat ) )
         => ( groups2027974829824023292at_nat(G3,sup_su4489774667511045786d_enat(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2027974829824023292at_nat(G3,A3)),groups2027974829824023292at_nat(G3,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8433_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_complex,B2: set_complex,G3: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),inf_inf_set_complex(A3,B2)))
             => ( aa_complex_int(G3,X5) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G3,sup_sup_set_complex(A3,B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G3,A3)),groups5690904116761175830ex_int(G3,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8434_sum_OatLeastAtMost__rev,axiom,
    ! [G3: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G3),set_or1269000886237332187st_nat(N,M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_oo(G3),N),M)),set_or1269000886237332187st_nat(N,M)) ).

% sum.atLeastAtMost_rev
tff(fact_8435_sum_OatLeastAtMost__rev,axiom,
    ! [G3: fun_nat_real,N: nat,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(G3),set_or1269000886237332187st_nat(N,M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_op(G3),N),M)),set_or1269000886237332187st_nat(N,M)) ).

% sum.atLeastAtMost_rev
tff(fact_8436_sum_Oremove,axiom,
    ! [A3: set_complex,X: complex,G3: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(member_complex(X),A3))
       => ( groups5808333547571424918x_real(G3,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G3,X)),groups5808333547571424918x_real(G3,minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_8437_sum_Oremove,axiom,
    ! [A3: set_Extended_enat,X: extended_enat,G3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A3))
       => ( groups4148127829035722712t_real(G3,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Ext2541572767299617110t_real(G3,X)),groups4148127829035722712t_real(G3,minus_925952699566721837d_enat(A3,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% sum.remove
tff(fact_8438_sum_Oremove,axiom,
    ! [A3: set_complex,X: complex,G3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(member_complex(X),A3))
       => ( groups5058264527183730370ex_rat(G3,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(G3,X)),groups5058264527183730370ex_rat(G3,minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_8439_sum_Oremove,axiom,
    ! [A3: set_Extended_enat,X: extended_enat,G3: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A3))
       => ( groups1392844769737527556at_rat(G3,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Extended_enat_rat(G3,X)),groups1392844769737527556at_rat(G3,minus_925952699566721837d_enat(A3,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% sum.remove
tff(fact_8440_sum_Oremove,axiom,
    ! [A3: set_complex,X: complex,G3: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(member_complex(X),A3))
       => ( groups5693394587270226106ex_nat(G3,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G3,X)),groups5693394587270226106ex_nat(G3,minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_8441_sum_Oremove,axiom,
    ! [A3: set_Extended_enat,X: extended_enat,G3: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A3))
       => ( groups2027974829824023292at_nat(G3,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Extended_enat_nat(G3,X)),groups2027974829824023292at_nat(G3,minus_925952699566721837d_enat(A3,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% sum.remove
tff(fact_8442_sum_Oremove,axiom,
    ! [A3: set_complex,X: complex,G3: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(member_complex(X),A3))
       => ( groups5690904116761175830ex_int(G3,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G3,X)),groups5690904116761175830ex_int(G3,minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_8443_sum_Oremove,axiom,
    ! [A3: set_Extended_enat,X: extended_enat,G3: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X),A3))
       => ( groups2025484359314973016at_int(G3,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Extended_enat_int(G3,X)),groups2025484359314973016at_int(G3,minus_925952699566721837d_enat(A3,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% sum.remove
tff(fact_8444_sum_Oremove,axiom,
    ! [A3: set_real,X: real,G3: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(member_real(X),A3))
       => ( groups8097168146408367636l_real(G3,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G3,X)),groups8097168146408367636l_real(G3,minus_minus_set_real(A3,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_8445_sum_Oremove,axiom,
    ! [A3: set_real,X: real,G3: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(member_real(X),A3))
       => ( groups1300246762558778688al_rat(G3,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(G3,X)),groups1300246762558778688al_rat(G3,minus_minus_set_real(A3,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_8446_sum_Oinsert__remove,axiom,
    ! [A3: set_complex,G3: fun_complex_real,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5808333547571424918x_real(G3,insert_complex(X,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G3,X)),groups5808333547571424918x_real(G3,minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_8447_sum_Oinsert__remove,axiom,
    ! [A3: set_Extended_enat,G3: fun_Ex1451620480498224165t_real,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups4148127829035722712t_real(G3,insert_Extended_enat(X,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Ext2541572767299617110t_real(G3,X)),groups4148127829035722712t_real(G3,minus_925952699566721837d_enat(A3,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ).

% sum.insert_remove
tff(fact_8448_sum_Oinsert__remove,axiom,
    ! [A3: set_complex,G3: fun_complex_rat,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5058264527183730370ex_rat(G3,insert_complex(X,A3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(G3,X)),groups5058264527183730370ex_rat(G3,minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_8449_sum_Oinsert__remove,axiom,
    ! [A3: set_Extended_enat,G3: fun_Ex4152069839811072081at_rat,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups1392844769737527556at_rat(G3,insert_Extended_enat(X,A3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Extended_enat_rat(G3,X)),groups1392844769737527556at_rat(G3,minus_925952699566721837d_enat(A3,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ).

% sum.insert_remove
tff(fact_8450_sum_Oinsert__remove,axiom,
    ! [A3: set_complex,G3: fun_complex_nat,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5693394587270226106ex_nat(G3,insert_complex(X,A3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G3,X)),groups5693394587270226106ex_nat(G3,minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_8451_sum_Oinsert__remove,axiom,
    ! [A3: set_Extended_enat,G3: fun_Ex7660298398204468809at_nat,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups2027974829824023292at_nat(G3,insert_Extended_enat(X,A3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Extended_enat_nat(G3,X)),groups2027974829824023292at_nat(G3,minus_925952699566721837d_enat(A3,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ).

% sum.insert_remove
tff(fact_8452_sum_Oinsert__remove,axiom,
    ! [A3: set_complex,G3: fun_complex_int,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5690904116761175830ex_int(G3,insert_complex(X,A3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G3,X)),groups5690904116761175830ex_int(G3,minus_811609699411566653omplex(A3,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_8453_sum_Oinsert__remove,axiom,
    ! [A3: set_Extended_enat,G3: fun_Ex3482447378695272101at_int,X: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups2025484359314973016at_int(G3,insert_Extended_enat(X,A3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Extended_enat_int(G3,X)),groups2025484359314973016at_int(G3,minus_925952699566721837d_enat(A3,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))) ) ) ).

% sum.insert_remove
tff(fact_8454_sum_Oinsert__remove,axiom,
    ! [A3: set_real,G3: fun_real_real,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups8097168146408367636l_real(G3,insert_real(X,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G3,X)),groups8097168146408367636l_real(G3,minus_minus_set_real(A3,insert_real(X,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_8455_sum_Oinsert__remove,axiom,
    ! [A3: set_real,G3: fun_real_rat,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups1300246762558778688al_rat(G3,insert_real(X,A3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(G3,X)),groups1300246762558778688al_rat(G3,minus_minus_set_real(A3,insert_real(X,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_8456_sum__diff1,axiom,
    ! [A3: set_complex,A2: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),A3))
         => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A3,insert_complex(A2,bot_bot_set_complex))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A3)),aa_complex_real(F,A2)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),A3))
         => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A3,insert_complex(A2,bot_bot_set_complex))) = groups5808333547571424918x_real(F,A3) ) ) ) ) ).

% sum_diff1
tff(fact_8457_sum__diff1,axiom,
    ! [A3: set_Extended_enat,A2: extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),A3))
         => ( groups4148127829035722712t_real(F,minus_925952699566721837d_enat(A3,insert_Extended_enat(A2,bot_bo7653980558646680370d_enat))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups4148127829035722712t_real(F,A3)),aa_Ext2541572767299617110t_real(F,A2)) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),A3))
         => ( groups4148127829035722712t_real(F,minus_925952699566721837d_enat(A3,insert_Extended_enat(A2,bot_bo7653980558646680370d_enat))) = groups4148127829035722712t_real(F,A3) ) ) ) ) ).

% sum_diff1
tff(fact_8458_sum__diff1,axiom,
    ! [A3: set_real,A2: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ( pp(aa_set_real_bool(member_real(A2),A3))
         => ( groups8097168146408367636l_real(F,minus_minus_set_real(A3,insert_real(A2,bot_bot_set_real))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8097168146408367636l_real(F,A3)),aa_real_real(F,A2)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),A3))
         => ( groups8097168146408367636l_real(F,minus_minus_set_real(A3,insert_real(A2,bot_bot_set_real))) = groups8097168146408367636l_real(F,A3) ) ) ) ) ).

% sum_diff1
tff(fact_8459_sum__diff1,axiom,
    ! [A3: set_bool,A2: bool,F: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( ( pp(aa_set_bool_bool(member_bool(A2),A3))
         => ( groups502887708668150281l_real(F,minus_minus_set_bool(A3,insert_bool(A2,bot_bot_set_bool))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups502887708668150281l_real(F,A3)),aa_bool_real(F,A2)) ) )
        & ( ~ pp(aa_set_bool_bool(member_bool(A2),A3))
         => ( groups502887708668150281l_real(F,minus_minus_set_bool(A3,insert_bool(A2,bot_bot_set_bool))) = groups502887708668150281l_real(F,A3) ) ) ) ) ).

% sum_diff1
tff(fact_8460_sum__diff1,axiom,
    ! [A3: set_int,A2: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( pp(aa_set_int_bool(member_int(A2),A3))
         => ( groups8778361861064173332t_real(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(A2,bot_bot_set_int))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A3)),aa_int_real(F,A2)) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A2),A3))
         => ( groups8778361861064173332t_real(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(A2,bot_bot_set_int))) = groups8778361861064173332t_real(F,A3) ) ) ) ) ).

% sum_diff1
tff(fact_8461_sum__diff1,axiom,
    ! [A3: set_complex,A2: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),A3))
         => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A3,insert_complex(A2,bot_bot_set_complex))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups5058264527183730370ex_rat(F,A3)),aa_complex_rat(F,A2)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),A3))
         => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A3,insert_complex(A2,bot_bot_set_complex))) = groups5058264527183730370ex_rat(F,A3) ) ) ) ) ).

% sum_diff1
tff(fact_8462_sum__diff1,axiom,
    ! [A3: set_Extended_enat,A2: extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),A3))
         => ( groups1392844769737527556at_rat(F,minus_925952699566721837d_enat(A3,insert_Extended_enat(A2,bot_bo7653980558646680370d_enat))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups1392844769737527556at_rat(F,A3)),aa_Extended_enat_rat(F,A2)) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),A3))
         => ( groups1392844769737527556at_rat(F,minus_925952699566721837d_enat(A3,insert_Extended_enat(A2,bot_bo7653980558646680370d_enat))) = groups1392844769737527556at_rat(F,A3) ) ) ) ) ).

% sum_diff1
tff(fact_8463_sum__diff1,axiom,
    ! [A3: set_real,A2: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ( pp(aa_set_real_bool(member_real(A2),A3))
         => ( groups1300246762558778688al_rat(F,minus_minus_set_real(A3,insert_real(A2,bot_bot_set_real))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups1300246762558778688al_rat(F,A3)),aa_real_rat(F,A2)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),A3))
         => ( groups1300246762558778688al_rat(F,minus_minus_set_real(A3,insert_real(A2,bot_bot_set_real))) = groups1300246762558778688al_rat(F,A3) ) ) ) ) ).

% sum_diff1
tff(fact_8464_sum__diff1,axiom,
    ! [A3: set_bool,A2: bool,F: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A3))
     => ( ( pp(aa_set_bool_bool(member_bool(A2),A3))
         => ( groups5133371213642249141ol_rat(F,minus_minus_set_bool(A3,insert_bool(A2,bot_bot_set_bool))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups5133371213642249141ol_rat(F,A3)),aa_bool_rat(F,A2)) ) )
        & ( ~ pp(aa_set_bool_bool(member_bool(A2),A3))
         => ( groups5133371213642249141ol_rat(F,minus_minus_set_bool(A3,insert_bool(A2,bot_bot_set_bool))) = groups5133371213642249141ol_rat(F,A3) ) ) ) ) ).

% sum_diff1
tff(fact_8465_sum__diff1,axiom,
    ! [A3: set_int,A2: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( pp(aa_set_int_bool(member_int(A2),A3))
         => ( groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(A2,bot_bot_set_int))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups3906332499630173760nt_rat(F,A3)),aa_int_rat(F,A2)) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A2),A3))
         => ( groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(A2,bot_bot_set_int))) = groups3906332499630173760nt_rat(F,A3) ) ) ) ) ).

% sum_diff1
tff(fact_8466_sum__Un,axiom,
    ! [A3: set_int,B2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups8778361861064173332t_real(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(F,A3)),groups8778361861064173332t_real(F,B2))),groups8778361861064173332t_real(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))) ) ) ) ).

% sum_Un
tff(fact_8467_sum__Un,axiom,
    ! [A3: set_complex,B2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups5808333547571424918x_real(F,sup_sup_set_complex(A3,B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(F,A3)),groups5808333547571424918x_real(F,B2))),groups5808333547571424918x_real(F,inf_inf_set_complex(A3,B2))) ) ) ) ).

% sum_Un
tff(fact_8468_sum__Un,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( groups4148127829035722712t_real(F,sup_su4489774667511045786d_enat(A3,B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups4148127829035722712t_real(F,A3)),groups4148127829035722712t_real(F,B2))),groups4148127829035722712t_real(F,inf_in8357106775501769908d_enat(A3,B2))) ) ) ) ).

% sum_Un
tff(fact_8469_sum__Un,axiom,
    ! [A3: set_int,B2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(F,B2))),groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))) ) ) ) ).

% sum_Un
tff(fact_8470_sum__Un,axiom,
    ! [A3: set_complex,B2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups5058264527183730370ex_rat(F,sup_sup_set_complex(A3,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(F,B2))),groups5058264527183730370ex_rat(F,inf_inf_set_complex(A3,B2))) ) ) ) ).

% sum_Un
tff(fact_8471_sum__Un,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( groups1392844769737527556at_rat(F,sup_su4489774667511045786d_enat(A3,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups1392844769737527556at_rat(F,A3)),groups1392844769737527556at_rat(F,B2))),groups1392844769737527556at_rat(F,inf_in8357106775501769908d_enat(A3,B2))) ) ) ) ).

% sum_Un
tff(fact_8472_sum__Un,axiom,
    ! [A3: set_nat,B2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups2906978787729119204at_rat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(F,A3)),groups2906978787729119204at_rat(F,B2))),groups2906978787729119204at_rat(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2))) ) ) ) ).

% sum_Un
tff(fact_8473_sum__Un,axiom,
    ! [A3: set_complex,B2: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups5690904116761175830ex_int(F,sup_sup_set_complex(A3,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(F,A3)),groups5690904116761175830ex_int(F,B2))),groups5690904116761175830ex_int(F,inf_inf_set_complex(A3,B2))) ) ) ) ).

% sum_Un
tff(fact_8474_sum__Un,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( groups2025484359314973016at_int(F,sup_su4489774667511045786d_enat(A3,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups2025484359314973016at_int(F,A3)),groups2025484359314973016at_int(F,B2))),groups2025484359314973016at_int(F,inf_in8357106775501769908d_enat(A3,B2))) ) ) ) ).

% sum_Un
tff(fact_8475_sum__Un,axiom,
    ! [A3: set_nat,B2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups3539618377306564664at_int(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(F,A3)),groups3539618377306564664at_int(F,B2))),groups3539618377306564664at_int(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2))) ) ) ) ).

% sum_Un
tff(fact_8476_sum_Ounion__disjoint,axiom,
    ! [A3: set_complex,B2: set_complex,G3: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ( inf_inf_set_complex(A3,B2) = bot_bot_set_complex )
         => ( groups5808333547571424918x_real(G3,sup_sup_set_complex(A3,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G3,A3)),groups5808333547571424918x_real(G3,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8477_sum_Ounion__disjoint,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( ( inf_in8357106775501769908d_enat(A3,B2) = bot_bo7653980558646680370d_enat )
         => ( groups4148127829035722712t_real(G3,sup_su4489774667511045786d_enat(A3,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups4148127829035722712t_real(G3,A3)),groups4148127829035722712t_real(G3,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8478_sum_Ounion__disjoint,axiom,
    ! [A3: set_complex,B2: set_complex,G3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ( inf_inf_set_complex(A3,B2) = bot_bot_set_complex )
         => ( groups5058264527183730370ex_rat(G3,sup_sup_set_complex(A3,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(G3,A3)),groups5058264527183730370ex_rat(G3,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8479_sum_Ounion__disjoint,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( ( inf_in8357106775501769908d_enat(A3,B2) = bot_bo7653980558646680370d_enat )
         => ( groups1392844769737527556at_rat(G3,sup_su4489774667511045786d_enat(A3,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups1392844769737527556at_rat(G3,A3)),groups1392844769737527556at_rat(G3,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8480_sum_Ounion__disjoint,axiom,
    ! [A3: set_complex,B2: set_complex,G3: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ( inf_inf_set_complex(A3,B2) = bot_bot_set_complex )
         => ( groups5693394587270226106ex_nat(G3,sup_sup_set_complex(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G3,A3)),groups5693394587270226106ex_nat(G3,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8481_sum_Ounion__disjoint,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( ( inf_in8357106775501769908d_enat(A3,B2) = bot_bo7653980558646680370d_enat )
         => ( groups2027974829824023292at_nat(G3,sup_su4489774667511045786d_enat(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2027974829824023292at_nat(G3,A3)),groups2027974829824023292at_nat(G3,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8482_sum_Ounion__disjoint,axiom,
    ! [A3: set_complex,B2: set_complex,G3: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ( inf_inf_set_complex(A3,B2) = bot_bot_set_complex )
         => ( groups5690904116761175830ex_int(G3,sup_sup_set_complex(A3,B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G3,A3)),groups5690904116761175830ex_int(G3,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8483_sum_Ounion__disjoint,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( ( inf_in8357106775501769908d_enat(A3,B2) = bot_bo7653980558646680370d_enat )
         => ( groups2025484359314973016at_int(G3,sup_su4489774667511045786d_enat(A3,B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups2025484359314973016at_int(G3,A3)),groups2025484359314973016at_int(G3,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8484_sum_Ounion__disjoint,axiom,
    ! [A3: set_real,B2: set_real,G3: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(finite_finite_real,B2))
       => ( ( inf_inf_set_real(A3,B2) = bot_bot_set_real )
         => ( groups8097168146408367636l_real(G3,sup_sup_set_real(A3,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8097168146408367636l_real(G3,A3)),groups8097168146408367636l_real(G3,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8485_sum_Ounion__disjoint,axiom,
    ! [A3: set_real,B2: set_real,G3: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(finite_finite_real,B2))
       => ( ( inf_inf_set_real(A3,B2) = bot_bot_set_real )
         => ( groups1300246762558778688al_rat(G3,sup_sup_set_real(A3,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups1300246762558778688al_rat(G3,A3)),groups1300246762558778688al_rat(G3,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8486_sum_Ounion__diff2,axiom,
    ! [A3: set_int,B2: set_int,G3: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups8778361861064173332t_real(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2))),groups8778361861064173332t_real(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B2),A3)))),groups8778361861064173332t_real(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8487_sum_Ounion__diff2,axiom,
    ! [A3: set_complex,B2: set_complex,G3: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups5808333547571424918x_real(G3,sup_sup_set_complex(A3,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G3,minus_811609699411566653omplex(A3,B2))),groups5808333547571424918x_real(G3,minus_811609699411566653omplex(B2,A3)))),groups5808333547571424918x_real(G3,inf_inf_set_complex(A3,B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8488_sum_Ounion__diff2,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( groups4148127829035722712t_real(G3,sup_su4489774667511045786d_enat(A3,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups4148127829035722712t_real(G3,minus_925952699566721837d_enat(A3,B2))),groups4148127829035722712t_real(G3,minus_925952699566721837d_enat(B2,A3)))),groups4148127829035722712t_real(G3,inf_in8357106775501769908d_enat(A3,B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8489_sum_Ounion__diff2,axiom,
    ! [A3: set_int,B2: set_int,G3: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups3906332499630173760nt_rat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2))),groups3906332499630173760nt_rat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B2),A3)))),groups3906332499630173760nt_rat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8490_sum_Ounion__diff2,axiom,
    ! [A3: set_complex,B2: set_complex,G3: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups5058264527183730370ex_rat(G3,sup_sup_set_complex(A3,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(G3,minus_811609699411566653omplex(A3,B2))),groups5058264527183730370ex_rat(G3,minus_811609699411566653omplex(B2,A3)))),groups5058264527183730370ex_rat(G3,inf_inf_set_complex(A3,B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8491_sum_Ounion__diff2,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( groups1392844769737527556at_rat(G3,sup_su4489774667511045786d_enat(A3,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups1392844769737527556at_rat(G3,minus_925952699566721837d_enat(A3,B2))),groups1392844769737527556at_rat(G3,minus_925952699566721837d_enat(B2,A3)))),groups1392844769737527556at_rat(G3,inf_in8357106775501769908d_enat(A3,B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8492_sum_Ounion__diff2,axiom,
    ! [A3: set_int,B2: set_int,G3: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups4541462559716669496nt_nat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2))),groups4541462559716669496nt_nat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B2),A3)))),groups4541462559716669496nt_nat(G3,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8493_sum_Ounion__diff2,axiom,
    ! [A3: set_complex,B2: set_complex,G3: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups5693394587270226106ex_nat(G3,sup_sup_set_complex(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G3,minus_811609699411566653omplex(A3,B2))),groups5693394587270226106ex_nat(G3,minus_811609699411566653omplex(B2,A3)))),groups5693394587270226106ex_nat(G3,inf_inf_set_complex(A3,B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8494_sum_Ounion__diff2,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( groups2027974829824023292at_nat(G3,sup_su4489774667511045786d_enat(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2027974829824023292at_nat(G3,minus_925952699566721837d_enat(A3,B2))),groups2027974829824023292at_nat(G3,minus_925952699566721837d_enat(B2,A3)))),groups2027974829824023292at_nat(G3,inf_in8357106775501769908d_enat(A3,B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8495_sum_Ounion__diff2,axiom,
    ! [A3: set_complex,B2: set_complex,G3: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups5690904116761175830ex_int(G3,sup_sup_set_complex(A3,B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G3,minus_811609699411566653omplex(A3,B2))),groups5690904116761175830ex_int(G3,minus_811609699411566653omplex(B2,A3)))),groups5690904116761175830ex_int(G3,inf_inf_set_complex(A3,B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8496_sum__Un2,axiom,
    ! [A3: set_int,B2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)))
     => ( groups8778361861064173332t_real(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2))),groups8778361861064173332t_real(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B2),A3)))),groups8778361861064173332t_real(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))) ) ) ).

% sum_Un2
tff(fact_8497_sum__Un2,axiom,
    ! [A3: set_complex,B2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,sup_sup_set_complex(A3,B2)))
     => ( groups5808333547571424918x_real(F,sup_sup_set_complex(A3,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(F,minus_811609699411566653omplex(A3,B2))),groups5808333547571424918x_real(F,minus_811609699411566653omplex(B2,A3)))),groups5808333547571424918x_real(F,inf_inf_set_complex(A3,B2))) ) ) ).

% sum_Un2
tff(fact_8498_sum__Un2,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,sup_su4489774667511045786d_enat(A3,B2)))
     => ( groups4148127829035722712t_real(F,sup_su4489774667511045786d_enat(A3,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups4148127829035722712t_real(F,minus_925952699566721837d_enat(A3,B2))),groups4148127829035722712t_real(F,minus_925952699566721837d_enat(B2,A3)))),groups4148127829035722712t_real(F,inf_in8357106775501769908d_enat(A3,B2))) ) ) ).

% sum_Un2
tff(fact_8499_sum__Un2,axiom,
    ! [A3: set_int,B2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)))
     => ( groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2))),groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B2),A3)))),groups3906332499630173760nt_rat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))) ) ) ).

% sum_Un2
tff(fact_8500_sum__Un2,axiom,
    ! [A3: set_complex,B2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,sup_sup_set_complex(A3,B2)))
     => ( groups5058264527183730370ex_rat(F,sup_sup_set_complex(A3,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A3,B2))),groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(B2,A3)))),groups5058264527183730370ex_rat(F,inf_inf_set_complex(A3,B2))) ) ) ).

% sum_Un2
tff(fact_8501_sum__Un2,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,sup_su4489774667511045786d_enat(A3,B2)))
     => ( groups1392844769737527556at_rat(F,sup_su4489774667511045786d_enat(A3,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups1392844769737527556at_rat(F,minus_925952699566721837d_enat(A3,B2))),groups1392844769737527556at_rat(F,minus_925952699566721837d_enat(B2,A3)))),groups1392844769737527556at_rat(F,inf_in8357106775501769908d_enat(A3,B2))) ) ) ).

% sum_Un2
tff(fact_8502_sum__Un2,axiom,
    ! [A3: set_int,B2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)))
     => ( groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2))),groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B2),A3)))),groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))) ) ) ).

% sum_Un2
tff(fact_8503_sum__Un2,axiom,
    ! [A3: set_complex,B2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,sup_sup_set_complex(A3,B2)))
     => ( groups5693394587270226106ex_nat(F,sup_sup_set_complex(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A3,B2))),groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(B2,A3)))),groups5693394587270226106ex_nat(F,inf_inf_set_complex(A3,B2))) ) ) ).

% sum_Un2
tff(fact_8504_sum__Un2,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,sup_su4489774667511045786d_enat(A3,B2)))
     => ( groups2027974829824023292at_nat(F,sup_su4489774667511045786d_enat(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2027974829824023292at_nat(F,minus_925952699566721837d_enat(A3,B2))),groups2027974829824023292at_nat(F,minus_925952699566721837d_enat(B2,A3)))),groups2027974829824023292at_nat(F,inf_in8357106775501769908d_enat(A3,B2))) ) ) ).

% sum_Un2
tff(fact_8505_sum__Un2,axiom,
    ! [A3: set_complex,B2: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,sup_sup_set_complex(A3,B2)))
     => ( groups5690904116761175830ex_int(F,sup_sup_set_complex(A3,B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A3,B2))),groups5690904116761175830ex_int(F,minus_811609699411566653omplex(B2,A3)))),groups5690904116761175830ex_int(F,inf_inf_set_complex(A3,B2))) ) ) ).

% sum_Un2
tff(fact_8506_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => ( suminf_int(F) = groups3539618377306564664at_int(F,N4) ) ) ) ).

% suminf_finite
tff(fact_8507_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => ( suminf_nat(F) = aa_set_nat_nat(groups3542108847815614940at_nat(F),N4) ) ) ) ).

% suminf_finite
tff(fact_8508_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => ( suminf_real(F) = aa_set_nat_real(groups6591440286371151544t_real(F),N4) ) ) ) ).

% suminf_finite
tff(fact_8509_sum_Odelta__remove,axiom,
    ! [S: set_complex,A2: complex,B3: fun_complex_real,C: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_oq(A2),B3),C),S) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(B3,A2)),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_oq(A2),B3),C),S) = groups5808333547571424918x_real(C,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8510_sum_Odelta__remove,axiom,
    ! [S: set_Extended_enat,A2: extended_enat,B3: fun_Ex1451620480498224165t_real,C: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),S))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aa_fun967516747568145260t_real(aTP_Lamm_or(A2),B3),C),S) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Ext2541572767299617110t_real(B3,A2)),groups4148127829035722712t_real(C,minus_925952699566721837d_enat(S,insert_Extended_enat(A2,bot_bo7653980558646680370d_enat)))) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),S))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aa_fun967516747568145260t_real(aTP_Lamm_or(A2),B3),C),S) = groups4148127829035722712t_real(C,minus_925952699566721837d_enat(S,insert_Extended_enat(A2,bot_bo7653980558646680370d_enat))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8511_sum_Odelta__remove,axiom,
    ! [S: set_complex,A2: complex,B3: fun_complex_rat,C: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_os(A2),B3),C),S) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(B3,A2)),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_os(A2),B3),C),S) = groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8512_sum_Odelta__remove,axiom,
    ! [S: set_Extended_enat,A2: extended_enat,B3: fun_Ex4152069839811072081at_rat,C: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),S))
         => ( groups1392844769737527556at_rat(aa_fun9057213682095532284at_rat(aa_fun4607285249076117784at_rat(aTP_Lamm_ot(A2),B3),C),S) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Extended_enat_rat(B3,A2)),groups1392844769737527556at_rat(C,minus_925952699566721837d_enat(S,insert_Extended_enat(A2,bot_bo7653980558646680370d_enat)))) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),S))
         => ( groups1392844769737527556at_rat(aa_fun9057213682095532284at_rat(aa_fun4607285249076117784at_rat(aTP_Lamm_ot(A2),B3),C),S) = groups1392844769737527556at_rat(C,minus_925952699566721837d_enat(S,insert_Extended_enat(A2,bot_bo7653980558646680370d_enat))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8513_sum_Odelta__remove,axiom,
    ! [S: set_complex,A2: complex,B3: fun_complex_nat,C: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_ou(A2),B3),C),S) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(B3,A2)),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_ou(A2),B3),C),S) = groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8514_sum_Odelta__remove,axiom,
    ! [S: set_Extended_enat,A2: extended_enat,B3: fun_Ex7660298398204468809at_nat,C: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),S))
         => ( groups2027974829824023292at_nat(aa_fun1810565495048290044at_nat(aa_fun5812849712608808208at_nat(aTP_Lamm_ov(A2),B3),C),S) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Extended_enat_nat(B3,A2)),groups2027974829824023292at_nat(C,minus_925952699566721837d_enat(S,insert_Extended_enat(A2,bot_bo7653980558646680370d_enat)))) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),S))
         => ( groups2027974829824023292at_nat(aa_fun1810565495048290044at_nat(aa_fun5812849712608808208at_nat(aTP_Lamm_ov(A2),B3),C),S) = groups2027974829824023292at_nat(C,minus_925952699566721837d_enat(S,insert_Extended_enat(A2,bot_bo7653980558646680370d_enat))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8515_sum_Odelta__remove,axiom,
    ! [S: set_complex,A2: complex,B3: fun_complex_int,C: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_ow(A2),B3),C),S) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(B3,A2)),groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_ow(A2),B3),C),S) = groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8516_sum_Odelta__remove,axiom,
    ! [S: set_Extended_enat,A2: extended_enat,B3: fun_Ex3482447378695272101at_int,C: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),S))
         => ( groups2025484359314973016at_int(aa_fun1695976788104708604at_int(aa_fun8556764835550362732at_int(aTP_Lamm_ox(A2),B3),C),S) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Extended_enat_int(B3,A2)),groups2025484359314973016at_int(C,minus_925952699566721837d_enat(S,insert_Extended_enat(A2,bot_bo7653980558646680370d_enat)))) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A2),S))
         => ( groups2025484359314973016at_int(aa_fun1695976788104708604at_int(aa_fun8556764835550362732at_int(aTP_Lamm_ox(A2),B3),C),S) = groups2025484359314973016at_int(C,minus_925952699566721837d_enat(S,insert_Extended_enat(A2,bot_bo7653980558646680370d_enat))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8517_sum_Odelta__remove,axiom,
    ! [S: set_real,A2: real,B3: fun_real_real,C: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_oy(A2),B3),C),S) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(B3,A2)),groups8097168146408367636l_real(C,minus_minus_set_real(S,insert_real(A2,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_oy(A2),B3),C),S) = groups8097168146408367636l_real(C,minus_minus_set_real(S,insert_real(A2,bot_bot_set_real))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8518_sum_Odelta__remove,axiom,
    ! [S: set_real,A2: real,B3: fun_real_rat,C: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_oz(A2),B3),C),S) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(B3,A2)),groups1300246762558778688al_rat(C,minus_minus_set_real(S,insert_real(A2,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_oz(A2),B3),C),S) = groups1300246762558778688al_rat(C,minus_minus_set_real(S,insert_real(A2,bot_bot_set_real))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8519_powser__sums__if,axiom,
    ! [M: nat,Z: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_pa(M),Z)),aa_nat_complex(power_power_complex(Z),M))) ).

% powser_sums_if
tff(fact_8520_powser__sums__if,axiom,
    ! [M: nat,Z: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_pb(M),Z)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M))) ).

% powser_sums_if
tff(fact_8521_powser__sums__if,axiom,
    ! [M: nat,Z: int] : pp(aa_int_bool(sums_int(aa_int_fun_nat_int(aTP_Lamm_pc(M),Z)),aa_nat_int(power_power_int(Z),M))) ).

% powser_sums_if
tff(fact_8522_powser__sums__zero,axiom,
    ! [A2: fun_nat_complex] : pp(aa_complex_bool(sums_complex(aTP_Lamm_gw(A2)),aa_nat_complex(A2,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_8523_powser__sums__zero,axiom,
    ! [A2: fun_nat_real] : pp(aa_real_bool(sums_real(aTP_Lamm_gx(A2)),aa_nat_real(A2,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_8524_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_rat,K: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8525_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_int,K: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8526_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_nat,K: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8527_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8528_sum_OatLeast0__atMost__Suc,axiom,
    ! [G3: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G3,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G3,set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_rat(G3,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8529_sum_OatLeast0__atMost__Suc,axiom,
    ! [G3: fun_nat_int,N: nat] : groups3539618377306564664at_int(G3,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G3,set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_int(G3,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8530_sum_OatLeast0__atMost__Suc,axiom,
    ! [G3: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G3),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G3),set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_nat(G3,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8531_sum_OatLeast0__atMost__Suc,axiom,
    ! [G3: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G3),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G3),set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_real(G3,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8532_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G3: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(G3,set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G3,M)),groups2906978787729119204at_rat(G3,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8533_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G3: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(G3,set_or1269000886237332187st_nat(M,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G3,M)),groups3539618377306564664at_int(G3,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8534_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G3: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G3),set_or1269000886237332187st_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G3,M)),aa_set_nat_nat(groups3542108847815614940at_nat(G3),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8535_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G3: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G3),set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G3,M)),aa_set_nat_real(groups6591440286371151544t_real(G3),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8536_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G3: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups2906978787729119204at_rat(G3,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G3,aa_nat_nat(suc,N))),groups2906978787729119204at_rat(G3,set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8537_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G3: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups3539618377306564664at_int(G3,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G3,aa_nat_nat(suc,N))),groups3539618377306564664at_int(G3,set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8538_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G3: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G3),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G3,aa_nat_nat(suc,N))),aa_set_nat_nat(groups3542108847815614940at_nat(G3),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8539_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G3: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G3),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G3,aa_nat_nat(suc,N))),aa_set_nat_real(groups6591440286371151544t_real(G3),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8540_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_8541_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_8542_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_8543_numeral__3__eq__3,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))) ).

% numeral_3_eq_3
tff(fact_8544_sum__strict__mono2,axiom,
    ! [B2: set_real,A3: set_real,B3: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2))
       => ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B2,A3)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,B3)))
           => ( ! [X5: real] :
                  ( pp(aa_set_real_bool(member_real(X5),B2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X5))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A3)),groups8097168146408367636l_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8545_sum__strict__mono2,axiom,
    ! [B2: set_bool,A3: set_bool,B3: bool,F: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,B2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),B2))
       => ( pp(aa_set_bool_bool(member_bool(B3),minus_minus_set_bool(B2,A3)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_bool_real(F,B3)))
           => ( ! [X5: bool] :
                  ( pp(aa_set_bool_bool(member_bool(X5),B2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_bool_real(F,X5))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups502887708668150281l_real(F,A3)),groups502887708668150281l_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8546_sum__strict__mono2,axiom,
    ! [B2: set_complex,A3: set_complex,B3: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),B2))
       => ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B2,A3)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,B3)))
           => ( ! [X5: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X5),B2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X5))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A3)),groups5808333547571424918x_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8547_sum__strict__mono2,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,B3: extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),B2))
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(B2,A3)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,B3)))
           => ( ! [X5: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),B2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,X5))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups4148127829035722712t_real(F,A3)),groups4148127829035722712t_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8548_sum__strict__mono2,axiom,
    ! [B2: set_real,A3: set_real,B3: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2))
       => ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B2,A3)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,B3)))
           => ( ! [X5: real] :
                  ( pp(aa_set_real_bool(member_real(X5),B2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X5))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1300246762558778688al_rat(F,A3)),groups1300246762558778688al_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8549_sum__strict__mono2,axiom,
    ! [B2: set_bool,A3: set_bool,B3: bool,F: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,B2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A3),B2))
       => ( pp(aa_set_bool_bool(member_bool(B3),minus_minus_set_bool(B2,A3)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_bool_rat(F,B3)))
           => ( ! [X5: bool] :
                  ( pp(aa_set_bool_bool(member_bool(X5),B2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_bool_rat(F,X5))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5133371213642249141ol_rat(F,A3)),groups5133371213642249141ol_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8550_sum__strict__mono2,axiom,
    ! [B2: set_complex,A3: set_complex,B3: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),B2))
       => ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B2,A3)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,B3)))
           => ( ! [X5: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X5),B2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X5))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8551_sum__strict__mono2,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,B3: extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),B2))
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(B2,A3)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_Extended_enat_rat(F,B3)))
           => ( ! [X5: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),B2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,X5))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1392844769737527556at_rat(F,A3)),groups1392844769737527556at_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8552_sum__strict__mono2,axiom,
    ! [B2: set_nat,A3: set_nat,B3: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),B2))
       => ( pp(aa_set_nat_bool(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A3)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,B3)))
           => ( ! [X5: nat] :
                  ( pp(aa_set_nat_bool(member_nat(X5),B2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X5))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A3)),groups2906978787729119204at_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8553_sum__strict__mono2,axiom,
    ! [B2: set_real,A3: set_real,B3: real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B2))
       => ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B2,A3)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,B3)))
           => ( ! [X5: real] :
                  ( pp(aa_set_real_bool(member_real(X5),B2))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X5))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups1935376822645274424al_nat(F,A3)),groups1935376822645274424al_nat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8554_member__le__sum,axiom,
    ! [I: complex,A3: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(member_complex(I),A3))
     => ( ! [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),minus_811609699411566653omplex(A3,insert_complex(I,bot_bot_set_complex))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X5))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I)),groups5808333547571424918x_real(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_8555_member__le__sum,axiom,
    ! [I: extended_enat,A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),A3))
     => ( ! [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),minus_925952699566721837d_enat(A3,insert_Extended_enat(I,bot_bo7653980558646680370d_enat))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,X5))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,I)),groups4148127829035722712t_real(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_8556_member__le__sum,axiom,
    ! [I: real,A3: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(member_real(I),A3))
     => ( ! [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),minus_minus_set_real(A3,insert_real(I,bot_bot_set_real))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X5))) )
       => ( pp(aa_set_real_bool(finite_finite_real,A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I)),groups8097168146408367636l_real(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_8557_member__le__sum,axiom,
    ! [I: bool,A3: set_bool,F: fun_bool_real] :
      ( pp(aa_set_bool_bool(member_bool(I),A3))
     => ( ! [X5: bool] :
            ( pp(aa_set_bool_bool(member_bool(X5),minus_minus_set_bool(A3,insert_bool(I,bot_bot_set_bool))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_bool_real(F,X5))) )
       => ( pp(aa_set_bool_bool(finite_finite_bool,A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_bool_real(F,I)),groups502887708668150281l_real(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_8558_member__le__sum,axiom,
    ! [I: int,A3: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(member_int(I),A3))
     => ( ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(I,bot_bot_set_int))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X5))) )
       => ( pp(aa_set_int_bool(finite_finite_int,A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I)),groups8778361861064173332t_real(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_8559_member__le__sum,axiom,
    ! [I: complex,A3: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(member_complex(I),A3))
     => ( ! [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),minus_811609699411566653omplex(A3,insert_complex(I,bot_bot_set_complex))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X5))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I)),groups5058264527183730370ex_rat(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_8560_member__le__sum,axiom,
    ! [I: extended_enat,A3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),A3))
     => ( ! [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),minus_925952699566721837d_enat(A3,insert_Extended_enat(I,bot_bo7653980558646680370d_enat))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,X5))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Extended_enat_rat(F,I)),groups1392844769737527556at_rat(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_8561_member__le__sum,axiom,
    ! [I: real,A3: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(member_real(I),A3))
     => ( ! [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),minus_minus_set_real(A3,insert_real(I,bot_bot_set_real))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X5))) )
       => ( pp(aa_set_real_bool(finite_finite_real,A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I)),groups1300246762558778688al_rat(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_8562_member__le__sum,axiom,
    ! [I: bool,A3: set_bool,F: fun_bool_rat] :
      ( pp(aa_set_bool_bool(member_bool(I),A3))
     => ( ! [X5: bool] :
            ( pp(aa_set_bool_bool(member_bool(X5),minus_minus_set_bool(A3,insert_bool(I,bot_bot_set_bool))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_bool_rat(F,X5))) )
       => ( pp(aa_set_bool_bool(finite_finite_bool,A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_bool_rat(F,I)),groups5133371213642249141ol_rat(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_8563_member__le__sum,axiom,
    ! [I: int,A3: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(member_int(I),A3))
     => ( ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(I,bot_bot_set_int))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X5))) )
       => ( pp(aa_set_int_bool(finite_finite_int,A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I)),groups3906332499630173760nt_rat(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_8564_sum__bounded__above__strict,axiom,
    ! [A3: set_real,F: fun_real_rat,K4: rat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,I2)),K4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_real(A3)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1300246762558778688al_rat(F,A3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_real(A3))),K4))) ) ) ).

% sum_bounded_above_strict
tff(fact_8565_sum__bounded__above__strict,axiom,
    ! [A3: set_bool,F: fun_bool_rat,K4: rat] :
      ( ! [I2: bool] :
          ( pp(aa_set_bool_bool(member_bool(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_bool_rat(F,I2)),K4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_bool(A3)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5133371213642249141ol_rat(F,A3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_bool(A3))),K4))) ) ) ).

% sum_bounded_above_strict
tff(fact_8566_sum__bounded__above__strict,axiom,
    ! [A3: set_complex,F: fun_complex_rat,K4: rat] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,I2)),K4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_complex(A3)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_complex(A3))),K4))) ) ) ).

% sum_bounded_above_strict
tff(fact_8567_sum__bounded__above__strict,axiom,
    ! [A3: set_nat,F: fun_nat_rat,K4: rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,I2)),K4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A3)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A3))),K4))) ) ) ).

% sum_bounded_above_strict
tff(fact_8568_sum__bounded__above__strict,axiom,
    ! [A3: set_int,F: fun_int_rat,K4: rat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,I2)),K4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_int(A3)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_int(A3))),K4))) ) ) ).

% sum_bounded_above_strict
tff(fact_8569_sum__bounded__above__strict,axiom,
    ! [A3: set_real,F: fun_real_nat,K4: nat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,I2)),K4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_real(A3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups1935376822645274424al_nat(F,A3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_real(A3))),K4))) ) ) ).

% sum_bounded_above_strict
tff(fact_8570_sum__bounded__above__strict,axiom,
    ! [A3: set_bool,F: fun_bool_nat,K4: nat] :
      ( ! [I2: bool] :
          ( pp(aa_set_bool_bool(member_bool(I2),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_bool_nat(F,I2)),K4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_bool(A3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5768501273728744877ol_nat(F,A3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_bool(A3))),K4))) ) ) ).

% sum_bounded_above_strict
tff(fact_8571_sum__bounded__above__strict,axiom,
    ! [A3: set_complex,F: fun_complex_nat,K4: nat] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,I2)),K4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_complex(A3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_complex(A3))),K4))) ) ) ).

% sum_bounded_above_strict
tff(fact_8572_sum__bounded__above__strict,axiom,
    ! [A3: set_int,F: fun_int_nat,K4: nat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,I2)),K4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_int(A3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_int(A3))),K4))) ) ) ).

% sum_bounded_above_strict
tff(fact_8573_sum__bounded__above__strict,axiom,
    ! [A3: set_real,F: fun_real_int,K4: int] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A3))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,I2)),K4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_real(A3)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups1932886352136224148al_int(F,A3)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_real(A3))),K4))) ) ) ).

% sum_bounded_above_strict
tff(fact_8574_sum__bounded__above__divide,axiom,
    ! [A3: set_complex,F: fun_complex_real,K4: real] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,K4),aa_nat_real(semiri5074537144036343181t_real,finite_card_complex(A3))))) )
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
       => ( ( A3 != bot_bot_set_complex )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,A3)),K4)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_8575_sum__bounded__above__divide,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real,K4: real] :
      ( ! [I2: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,I2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,K4),aa_nat_real(semiri5074537144036343181t_real,finite121521170596916366d_enat(A3))))) )
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
       => ( ( A3 != bot_bo7653980558646680370d_enat )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups4148127829035722712t_real(F,A3)),K4)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_8576_sum__bounded__above__divide,axiom,
    ! [A3: set_real,F: fun_real_real,K4: real] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,K4),aa_nat_real(semiri5074537144036343181t_real,finite_card_real(A3))))) )
     => ( pp(aa_set_real_bool(finite_finite_real,A3))
       => ( ( A3 != bot_bot_set_real )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A3)),K4)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_8577_sum__bounded__above__divide,axiom,
    ! [A3: set_bool,F: fun_bool_real,K4: real] :
      ( ! [I2: bool] :
          ( pp(aa_set_bool_bool(member_bool(I2),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_bool_real(F,I2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,K4),aa_nat_real(semiri5074537144036343181t_real,finite_card_bool(A3))))) )
     => ( pp(aa_set_bool_bool(finite_finite_bool,A3))
       => ( ( A3 != bot_bot_set_bool )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups502887708668150281l_real(F,A3)),K4)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_8578_sum__bounded__above__divide,axiom,
    ! [A3: set_int,F: fun_int_real,K4: real] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,K4),aa_nat_real(semiri5074537144036343181t_real,finite_card_int(A3))))) )
     => ( pp(aa_set_int_bool(finite_finite_int,A3))
       => ( ( A3 != bot_bot_set_int )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,A3)),K4)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_8579_sum__bounded__above__divide,axiom,
    ! [A3: set_complex,F: fun_complex_rat,K4: rat] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,K4),aa_nat_rat(semiri681578069525770553at_rat,finite_card_complex(A3))))) )
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
       => ( ( A3 != bot_bot_set_complex )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A3)),K4)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_8580_sum__bounded__above__divide,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat,K4: rat] :
      ( ! [I2: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Extended_enat_rat(F,I2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,K4),aa_nat_rat(semiri681578069525770553at_rat,finite121521170596916366d_enat(A3))))) )
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
       => ( ( A3 != bot_bo7653980558646680370d_enat )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1392844769737527556at_rat(F,A3)),K4)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_8581_sum__bounded__above__divide,axiom,
    ! [A3: set_real,F: fun_real_rat,K4: rat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,K4),aa_nat_rat(semiri681578069525770553at_rat,finite_card_real(A3))))) )
     => ( pp(aa_set_real_bool(finite_finite_real,A3))
       => ( ( A3 != bot_bot_set_real )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A3)),K4)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_8582_sum__bounded__above__divide,axiom,
    ! [A3: set_bool,F: fun_bool_rat,K4: rat] :
      ( ! [I2: bool] :
          ( pp(aa_set_bool_bool(member_bool(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_bool_rat(F,I2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,K4),aa_nat_rat(semiri681578069525770553at_rat,finite_card_bool(A3))))) )
     => ( pp(aa_set_bool_bool(finite_finite_bool,A3))
       => ( ( A3 != bot_bot_set_bool )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5133371213642249141ol_rat(F,A3)),K4)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_8583_sum__bounded__above__divide,axiom,
    ! [A3: set_nat,F: fun_nat_rat,K4: rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,K4),aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A3))))) )
     => ( pp(aa_set_nat_bool(finite_finite_nat,A3))
       => ( ( A3 != bot_bot_set_nat )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,A3)),K4)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_8584_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G3: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G3,set_or1269000886237332187st_nat(M,N))),aa_nat_rat(G3,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G3,M)),groups2906978787729119204at_rat(aTP_Lamm_pd(G3),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8585_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G3: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G3,set_or1269000886237332187st_nat(M,N))),aa_nat_int(G3,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G3,M)),groups3539618377306564664at_int(aTP_Lamm_ny(G3),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8586_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G3: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G3),set_or1269000886237332187st_nat(M,N))),aa_nat_nat(G3,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G3,M)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_nw(G3)),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8587_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G3: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G3),set_or1269000886237332187st_nat(M,N))),aa_nat_real(G3,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G3,M)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hv(G3)),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8588_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups2906978787729119204at_rat(aTP_Lamm_pe(F),set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_8589_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups3539618377306564664at_int(aTP_Lamm_pf(F),set_or1269000886237332187st_nat(M,N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_8590_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_pg(F)),set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,aa_nat_nat(suc,N))),aa_nat_real(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_8591_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_re6978948838062957830nteger,A2: fun_re6978948838062957830nteger,B3: code_integer,Delta: code_integer] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(X,I2))) )
     => ( ( groups7713935264441627589nteger(X,I5) = one_one_Code_integer )
       => ( ! [I2: real] :
              ( pp(aa_set_real_bool(member_real(I2),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_real_Code_integer(A2,I2),B3))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_ph(X),A2),I5),B3))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8592_convex__sum__bound__le,axiom,
    ! [I5: set_bool,X: fun_bo7282069421189217543nteger,A2: fun_bo7282069421189217543nteger,B3: code_integer,Delta: code_integer] :
      ( ! [I2: bool] :
          ( pp(aa_set_bool_bool(member_bool(I2),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_bool_Code_integer(X,I2))) )
     => ( ( groups1682264926906508858nteger(X,I5) = one_one_Code_integer )
       => ( ! [I2: bool] :
              ( pp(aa_set_bool_bool(member_bool(I2),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_bool_Code_integer(A2,I2),B3))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups1682264926906508858nteger(aa_fun1377107737837666940nteger(aTP_Lamm_pi(X),A2),I5),B3))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8593_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_Code_integer,A2: fun_nat_Code_integer,B3: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(X,I2))) )
     => ( ( groups7501900531339628137nteger(X,I5) = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( pp(aa_set_nat_bool(member_nat(I2),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_nat_Code_integer(A2,I2),B3))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_pj(X),A2),I5),B3))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8594_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_Code_integer,A2: fun_int_Code_integer,B3: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(X,I2))) )
     => ( ( groups7873554091576472773nteger(X,I5) = one_one_Code_integer )
       => ( ! [I2: int] :
              ( pp(aa_set_int_bool(member_int(I2),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_int_Code_integer(A2,I2),B3))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_pk(X),A2),I5),B3))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8595_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_real_real,A2: fun_real_real,B3: real,Delta: real] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(X,I2))) )
     => ( ( groups8097168146408367636l_real(X,I5) = one_one_real )
       => ( ! [I2: real] :
              ( pp(aa_set_real_bool(member_real(I2),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(A2,I2)),B3))),Delta)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_pl(X),A2),I5)),B3))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8596_convex__sum__bound__le,axiom,
    ! [I5: set_bool,X: fun_bool_real,A2: fun_bool_real,B3: real,Delta: real] :
      ( ! [I2: bool] :
          ( pp(aa_set_bool_bool(member_bool(I2),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_bool_real(X,I2))) )
     => ( ( groups502887708668150281l_real(X,I5) = one_one_real )
       => ( ! [I2: bool] :
              ( pp(aa_set_bool_bool(member_bool(I2),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_bool_real(A2,I2)),B3))),Delta)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups502887708668150281l_real(aa_fun8933258245593490108l_real(aTP_Lamm_pm(X),A2),I5)),B3))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8597_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_real,A2: fun_int_real,B3: real,Delta: real] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(X,I2))) )
     => ( ( groups8778361861064173332t_real(X,I5) = one_one_real )
       => ( ! [I2: int] :
              ( pp(aa_set_int_bool(member_int(I2),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(A2,I2)),B3))),Delta)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_pn(X),A2),I5)),B3))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8598_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_real_rat,A2: fun_real_rat,B3: rat,Delta: rat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),I5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(X,I2))) )
     => ( ( groups1300246762558778688al_rat(X,I5) = one_one_rat )
       => ( ! [I2: real] :
              ( pp(aa_set_real_bool(member_real(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_real_rat(A2,I2)),B3))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_po(X),A2),I5)),B3))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8599_convex__sum__bound__le,axiom,
    ! [I5: set_bool,X: fun_bool_rat,A2: fun_bool_rat,B3: rat,Delta: rat] :
      ( ! [I2: bool] :
          ( pp(aa_set_bool_bool(member_bool(I2),I5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_bool_rat(X,I2))) )
     => ( ( groups5133371213642249141ol_rat(X,I5) = one_one_rat )
       => ( ! [I2: bool] :
              ( pp(aa_set_bool_bool(member_bool(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_bool_rat(A2,I2)),B3))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups5133371213642249141ol_rat(aa_fun224000846028461884ol_rat(aTP_Lamm_pp(X),A2),I5)),B3))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8600_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_rat,A2: fun_nat_rat,B3: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),I5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(X,I2))) )
     => ( ( groups2906978787729119204at_rat(X,I5) = one_one_rat )
       => ( ! [I2: nat] :
              ( pp(aa_set_nat_bool(member_nat(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(A2,I2)),B3))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_pq(X),A2),I5)),B3))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8601_num_Osize_I6_J,axiom,
    ! [X32: num] : aa_num_nat(size_size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(6)
tff(fact_8602_sum__norm__bound,axiom,
    ! [S: set_real,F: fun_real_complex,K4: real] :
      ( ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(F,X5))),K4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups5754745047067104278omplex(F,S))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_real(S))),K4))) ) ).

% sum_norm_bound
tff(fact_8603_sum__norm__bound,axiom,
    ! [S: set_bool,F: fun_bool_complex,K4: real] :
      ( ! [X5: bool] :
          ( pp(aa_set_bool_bool(member_bool(X5),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_bool_complex(F,X5))),K4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups2959343415743557771omplex(F,S))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_bool(S))),K4))) ) ).

% sum_norm_bound
tff(fact_8604_sum__norm__bound,axiom,
    ! [S: set_list_nat,F: fun_list_nat_complex,K4: real] :
      ( ! [X5: list_nat] :
          ( member_list_nat(X5,S)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_list_nat_complex(F,X5))),K4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups6529277132148336714omplex(F,S))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_list_nat(S))),K4))) ) ).

% sum_norm_bound
tff(fact_8605_sum__norm__bound,axiom,
    ! [S: set_set_nat,F: fun_set_nat_complex,K4: real] :
      ( ! [X5: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X5),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(F,X5))),K4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups8255218700646806128omplex(F,S))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_set_nat(S))),K4))) ) ).

% sum_norm_bound
tff(fact_8606_sum__norm__bound,axiom,
    ! [S: set_nat,F: fun_nat_complex,K4: real] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,X5))),K4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),S))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(finite_card_nat,S))),K4))) ) ).

% sum_norm_bound
tff(fact_8607_sum__norm__bound,axiom,
    ! [S: set_int,F: fun_int_complex,K4: real] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(F,X5))),K4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups3049146728041665814omplex(F,S))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_int(S))),K4))) ) ).

% sum_norm_bound
tff(fact_8608_sum__norm__bound,axiom,
    ! [S: set_complex,F: fun_complex_complex,K4: real] :
      ( ! [X5: complex] :
          ( pp(aa_set_complex_bool(member_complex(X5),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(F,X5))),K4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,S))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_complex(S))),K4))) ) ).

% sum_norm_bound
tff(fact_8609_sum__norm__bound,axiom,
    ! [S: set_nat,F: fun_nat_real,K4: real] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,X5))),K4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),S))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(finite_card_nat,S))),K4))) ) ).

% sum_norm_bound
tff(fact_8610_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G3: fun_nat_rat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups2906978787729119204at_rat(G3,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G3,set_or1269000886237332187st_nat(M,N))),groups2906978787729119204at_rat(G3,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_8611_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G3: fun_nat_int,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups3539618377306564664at_int(G3,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G3,set_or1269000886237332187st_nat(M,N))),groups3539618377306564664at_int(G3,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_8612_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G3: fun_nat_nat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G3),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G3),set_or1269000886237332187st_nat(M,N))),aa_set_nat_nat(groups3542108847815614940at_nat(G3),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_8613_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G3: fun_nat_real,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G3),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G3),set_or1269000886237332187st_nat(M,N))),aa_set_nat_real(groups6591440286371151544t_real(G3),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_8614_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_8615_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_8616_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_8617_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_8618_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_8619_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_8620_sum__le__suminf,axiom,
    ! [F: fun_nat_int,I5: set_nat] :
      ( summable_int(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I5)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,I5)),suminf_int(F))) ) ) ) ).

% sum_le_suminf
tff(fact_8621_sum__le__suminf,axiom,
    ! [F: fun_nat_nat,I5: set_nat] :
      ( summable_nat(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I5)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),I5)),suminf_nat(F))) ) ) ) ).

% sum_le_suminf
tff(fact_8622_sum__le__suminf,axiom,
    ! [F: fun_nat_real,I5: set_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I5)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),I5)),suminf_real(F))) ) ) ) ).

% sum_le_suminf
tff(fact_8623_card__3__iff,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X3: product_prod_nat_nat,Y2: product_prod_nat_nat,Z2: product_prod_nat_nat] :
          ( ( S = insert8211810215607154385at_nat(X3,insert8211810215607154385at_nat(Y2,insert8211810215607154385at_nat(Z2,bot_bo2099793752762293965at_nat))) )
          & ( X3 != Y2 )
          & ( Y2 != Z2 )
          & ( X3 != Z2 ) ) ) ).

% card_3_iff
tff(fact_8624_card__3__iff,axiom,
    ! [S: set_complex] :
      ( ( finite_card_complex(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X3: complex,Y2: complex,Z2: complex] :
          ( ( S = insert_complex(X3,insert_complex(Y2,insert_complex(Z2,bot_bot_set_complex))) )
          & ( X3 != Y2 )
          & ( Y2 != Z2 )
          & ( X3 != Z2 ) ) ) ).

% card_3_iff
tff(fact_8625_card__3__iff,axiom,
    ! [S: set_list_nat] :
      ( ( finite_card_list_nat(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X3: list_nat,Y2: list_nat,Z2: list_nat] :
          ( ( S = insert_list_nat(X3,insert_list_nat(Y2,insert_list_nat(Z2,bot_bot_set_list_nat))) )
          & ( X3 != Y2 )
          & ( Y2 != Z2 )
          & ( X3 != Z2 ) ) ) ).

% card_3_iff
tff(fact_8626_card__3__iff,axiom,
    ! [S: set_set_nat] :
      ( ( finite_card_set_nat(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X3: set_nat,Y2: set_nat,Z2: set_nat] :
          ( ( S = insert_set_nat(X3,insert_set_nat(Y2,insert_set_nat(Z2,bot_bot_set_set_nat))) )
          & ( X3 != Y2 )
          & ( Y2 != Z2 )
          & ( X3 != Z2 ) ) ) ).

% card_3_iff
tff(fact_8627_card__3__iff,axiom,
    ! [S: set_real] :
      ( ( finite_card_real(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X3: real,Y2: real,Z2: real] :
          ( ( S = insert_real(X3,insert_real(Y2,insert_real(Z2,bot_bot_set_real))) )
          & ( X3 != Y2 )
          & ( Y2 != Z2 )
          & ( X3 != Z2 ) ) ) ).

% card_3_iff
tff(fact_8628_card__3__iff,axiom,
    ! [S: set_bool] :
      ( ( finite_card_bool(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X3: bool,Y2: bool,Z2: bool] :
          ( ( S = insert_bool(X3,insert_bool(Y2,insert_bool(Z2,bot_bot_set_bool))) )
          & ~ ( pp(X3)
            <=> pp(Y2) )
          & ~ ( pp(Y2)
            <=> pp(Z2) )
          & ~ ( pp(X3)
            <=> pp(Z2) ) ) ) ).

% card_3_iff
tff(fact_8629_card__3__iff,axiom,
    ! [S: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X3: nat,Y2: nat,Z2: nat] :
          ( ( S = insert_nat(X3,insert_nat(Y2,insert_nat(Z2,bot_bot_set_nat))) )
          & ( X3 != Y2 )
          & ( Y2 != Z2 )
          & ( X3 != Z2 ) ) ) ).

% card_3_iff
tff(fact_8630_card__3__iff,axiom,
    ! [S: set_int] :
      ( ( finite_card_int(S) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X3: int,Y2: int,Z2: int] :
          ( ( S = insert_int(X3,insert_int(Y2,insert_int(Z2,bot_bot_set_int))) )
          & ( X3 != Y2 )
          & ( Y2 != Z2 )
          & ( X3 != Z2 ) ) ) ).

% card_3_iff
tff(fact_8631_exp__le,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) ).

% exp_le
tff(fact_8632_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_8633_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups2906978787729119204at_rat(aTP_Lamm_pr(F),set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,M)),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups2906978787729119204at_rat(aTP_Lamm_pr(F),set_or1269000886237332187st_nat(M,N)) = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
tff(fact_8634_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups3539618377306564664at_int(aTP_Lamm_ps(F),set_or1269000886237332187st_nat(M,N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups3539618377306564664at_int(aTP_Lamm_ps(F),set_or1269000886237332187st_nat(M,N)) = zero_zero_int ) ) ) ).

% sum_natinterval_diff
tff(fact_8635_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_pt(F)),set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_pt(F)),set_or1269000886237332187st_nat(M,N)) = zero_zero_real ) ) ) ).

% sum_natinterval_diff
tff(fact_8636_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(aTP_Lamm_pu(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,N)),aa_nat_rat(F,M)) ) ) ).

% sum_telescope''
tff(fact_8637_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(aTP_Lamm_pv(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,N)),aa_nat_int(F,M)) ) ) ).

% sum_telescope''
tff(fact_8638_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_pw(F)),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,N)),aa_nat_real(F,M)) ) ) ).

% sum_telescope''
tff(fact_8639_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_complex,E2: real] :
      ( summable_complex(F)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M2))
                 => ! [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),set_or1269000886237332187st_nat(M2,N5)))),E2)) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8640_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_real,E2: real] :
      ( summable_real(F)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M2))
                 => ! [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(M2,N5)))),E2)) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8641_geometric__sums,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
     => pp(aa_real_bool(sums_real(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)))) ) ).

% geometric_sums
tff(fact_8642_geometric__sums,axiom,
    ! [C: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
     => pp(aa_complex_bool(sums_complex(power_power_complex(C)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C)))) ) ).

% geometric_sums
tff(fact_8643_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) = groups3539618377306564664at_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bk,N))) ).

% mask_eq_sum_exp
tff(fact_8644_mask__eq__sum__exp,axiom,
    ! [N: nat] : minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N),one_one_Code_integer) = groups7501900531339628137nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bk,N))) ).

% mask_eq_sum_exp
tff(fact_8645_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bk,N))) ).

% mask_eq_sum_exp
tff(fact_8646_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),set_or1269000886237332187st_nat(M,N))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_8647_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)),groups2906978787729119204at_rat(power_power_rat(X),set_or1269000886237332187st_nat(M,N))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),M)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_8648_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),groups3539618377306564664at_int(power_power_int(X),set_or1269000886237332187st_nat(M,N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),M)),aa_nat_int(power_power_int(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_8649_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),set_or1269000886237332187st_nat(M,N))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_8650_sum_Oin__pairs,axiom,
    ! [G3: fun_nat_rat,M: nat,N: nat] : groups2906978787729119204at_rat(G3,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups2906978787729119204at_rat(aTP_Lamm_px(G3),set_or1269000886237332187st_nat(M,N)) ).

% sum.in_pairs
tff(fact_8651_sum_Oin__pairs,axiom,
    ! [G3: fun_nat_int,M: nat,N: nat] : groups3539618377306564664at_int(G3,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups3539618377306564664at_int(aTP_Lamm_py(G3),set_or1269000886237332187st_nat(M,N)) ).

% sum.in_pairs
tff(fact_8652_sum_Oin__pairs,axiom,
    ! [G3: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G3),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pz(G3)),set_or1269000886237332187st_nat(M,N)) ).

% sum.in_pairs
tff(fact_8653_sum_Oin__pairs,axiom,
    ! [G3: fun_nat_real,M: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G3),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_qa(G3)),set_or1269000886237332187st_nat(M,N)) ).

% sum.in_pairs
tff(fact_8654_accp__subset__induct,axiom,
    ! [D4: fun_Pr2203583272059607673t_bool,R: fun_Pr1775001904083774893t_bool,X: produc4953844613479565601on_nat,P: fun_Pr2203583272059607673t_bool] :
      ( ord_le8032738712617130713t_bool(D4,accp_P8646395344606611882on_nat(R))
     => ( ! [X5: produc4953844613479565601on_nat,Z3: produc4953844613479565601on_nat] :
            ( pp(aa_Pro7647172495381830770t_bool(D4,X5))
           => ( pp(aa_Pro7647172495381830770t_bool(aa_Pro5001463314721565534t_bool(R,Z3),X5))
             => pp(aa_Pro7647172495381830770t_bool(D4,Z3)) ) )
       => ( pp(aa_Pro7647172495381830770t_bool(D4,X))
         => ( ! [X5: produc4953844613479565601on_nat] :
                ( pp(aa_Pro7647172495381830770t_bool(D4,X5))
               => ( ! [Z4: produc4953844613479565601on_nat] :
                      ( pp(aa_Pro7647172495381830770t_bool(aa_Pro5001463314721565534t_bool(R,Z4),X5))
                     => pp(aa_Pro7647172495381830770t_bool(P,Z4)) )
                 => pp(aa_Pro7647172495381830770t_bool(P,X5)) ) )
           => pp(aa_Pro7647172495381830770t_bool(P,X)) ) ) ) ) ).

% accp_subset_induct
tff(fact_8655_accp__subset__induct,axiom,
    ! [D4: fun_Pr7757163693017731481t_bool,R: fun_Pr248683899324676269t_bool,X: product_prod_nat_nat,P: fun_Pr7757163693017731481t_bool] :
      ( ord_le2468614981408567801t_bool(D4,accp_P4275260045618599050at_nat(R))
     => ( ! [X5: product_prod_nat_nat,Z3: product_prod_nat_nat] :
            ( pp(aa_Pro2232559814647311250t_bool(D4,X5))
           => ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,Z3),X5))
             => pp(aa_Pro2232559814647311250t_bool(D4,Z3)) ) )
       => ( pp(aa_Pro2232559814647311250t_bool(D4,X))
         => ( ! [X5: product_prod_nat_nat] :
                ( pp(aa_Pro2232559814647311250t_bool(D4,X5))
               => ( ! [Z4: product_prod_nat_nat] :
                      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,Z4),X5))
                     => pp(aa_Pro2232559814647311250t_bool(P,Z4)) )
                 => pp(aa_Pro2232559814647311250t_bool(P,X5)) ) )
           => pp(aa_Pro2232559814647311250t_bool(P,X)) ) ) ) ) ).

% accp_subset_induct
tff(fact_8656_accp__subset__induct,axiom,
    ! [D4: fun_Pr803113274115125473t_bool,R: fun_Pr3099820731930151597t_bool,X: product_prod_int_int,P: fun_Pr803113274115125473t_bool] :
      ( ord_le4737936599360737601t_bool(D4,accp_P1096762738010456898nt_int(R))
     => ( ! [X5: product_prod_int_int,Z3: product_prod_int_int] :
            ( pp(aa_Pro4501881432599481050t_bool(D4,X5))
           => ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(R,Z3),X5))
             => pp(aa_Pro4501881432599481050t_bool(D4,Z3)) ) )
       => ( pp(aa_Pro4501881432599481050t_bool(D4,X))
         => ( ! [X5: product_prod_int_int] :
                ( pp(aa_Pro4501881432599481050t_bool(D4,X5))
               => ( ! [Z4: product_prod_int_int] :
                      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(R,Z4),X5))
                     => pp(aa_Pro4501881432599481050t_bool(P,Z4)) )
                 => pp(aa_Pro4501881432599481050t_bool(P,X5)) ) )
           => pp(aa_Pro4501881432599481050t_bool(P,X)) ) ) ) ) ).

% accp_subset_induct
tff(fact_8657_accp__subset__induct,axiom,
    ! [D4: fun_list_nat_bool,R: fun_li8598321315282354825t_bool,X: list_nat,P: fun_list_nat_bool] :
      ( ord_le2936377143570524336t_bool(D4,accp_list_nat(R))
     => ( ! [X5: list_nat,Z3: list_nat] :
            ( pp(aa_list_nat_bool(D4,X5))
           => ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(R,Z3),X5))
             => pp(aa_list_nat_bool(D4,Z3)) ) )
       => ( pp(aa_list_nat_bool(D4,X))
         => ( ! [X5: list_nat] :
                ( pp(aa_list_nat_bool(D4,X5))
               => ( ! [Z4: list_nat] :
                      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(R,Z4),X5))
                     => pp(aa_list_nat_bool(P,Z4)) )
                 => pp(aa_list_nat_bool(P,X5)) ) )
           => pp(aa_list_nat_bool(P,X)) ) ) ) ) ).

% accp_subset_induct
tff(fact_8658_accp__subset__induct,axiom,
    ! [D4: fun_nat_bool,R: fun_nat_fun_nat_bool,X: nat,P: fun_nat_bool] :
      ( ord_le8840051076831633728t_bool(D4,accp_nat(R))
     => ( ! [X5: nat,Z3: nat] :
            ( pp(aa_nat_bool(D4,X5))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Z3),X5))
             => pp(aa_nat_bool(D4,Z3)) ) )
       => ( pp(aa_nat_bool(D4,X))
         => ( ! [X5: nat] :
                ( pp(aa_nat_bool(D4,X5))
               => ( ! [Z4: nat] :
                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Z4),X5))
                     => pp(aa_nat_bool(P,Z4)) )
                 => pp(aa_nat_bool(P,X5)) ) )
           => pp(aa_nat_bool(P,X)) ) ) ) ) ).

% accp_subset_induct
tff(fact_8659_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bk,N))) ).

% mask_eq_sum_exp_nat
tff(fact_8660_gauss__sum__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_qb),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_8661_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_qc(K)),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8662_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : groups2906978787729119204at_rat(aTP_Lamm_qd(K),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8663_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_qe(K)),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8664_double__arith__series,axiom,
    ! [A2: complex,D: complex,N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qf(A2),D)),set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A2)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),D))) ).

% double_arith_series
tff(fact_8665_double__arith__series,axiom,
    ! [A2: rat,D: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_qg(A2),D),set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),D))) ).

% double_arith_series
tff(fact_8666_double__arith__series,axiom,
    ! [A2: extended_enat,D: extended_enat,N: nat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one))),groups7108830773950497114d_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_qh(A2),D),set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)),one_on7984719198319812577d_enat)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one))),A2)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)),D))) ).

% double_arith_series
tff(fact_8667_double__arith__series,axiom,
    ! [A2: code_integer,D: code_integer,N: nat] : times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_qi(A2),D),set_or1269000886237332187st_nat(zero_zero_nat,N))) = times_3573771949741848930nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer),plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),A2),times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),D))) ).

% double_arith_series
tff(fact_8668_double__arith__series,axiom,
    ! [A2: int,D: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_qj(A2),D),set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D))) ).

% double_arith_series
tff(fact_8669_double__arith__series,axiom,
    ! [A2: nat,D: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qk(A2),D)),set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D))) ).

% double_arith_series
tff(fact_8670_double__arith__series,axiom,
    ! [A2: real,D: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ql(A2),D)),set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),D))) ).

% double_arith_series
tff(fact_8671_double__gauss__sum,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(semiri8010041392384452111omplex),set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).

% double_gauss_sum
tff(fact_8672_double__gauss__sum,axiom,
    ! [N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ).

% double_gauss_sum
tff(fact_8673_double__gauss__sum,axiom,
    ! [N: nat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one))),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)),one_on7984719198319812577d_enat)) ).

% double_gauss_sum
tff(fact_8674_double__gauss__sum,axiom,
    ! [N: nat] : times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(zero_zero_nat,N))) = times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer)) ).

% double_gauss_sum
tff(fact_8675_double__gauss__sum,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum
tff(fact_8676_double__gauss__sum,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ).

% double_gauss_sum
tff(fact_8677_double__gauss__sum,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(semiri5074537144036343181t_real),set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)) ).

% double_gauss_sum
tff(fact_8678_arith__series__nat,axiom,
    ! [A2: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qm(A2),D)),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_8679_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_qb),set_or1269000886237332187st_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_8680_arith__series,axiom,
    ! [A2: code_integer,D: code_integer,N: nat] : groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_qi(A2),D),set_or1269000886237332187st_nat(zero_zero_nat,N)) = divide6298287555418463151nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer),plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),A2),times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),D))),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% arith_series
tff(fact_8681_arith__series,axiom,
    ! [A2: int,D: int,N: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_qj(A2),D),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_8682_arith__series,axiom,
    ! [A2: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qk(A2),D)),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_8683_gauss__sum,axiom,
    ! [N: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(zero_zero_nat,N)) = divide6298287555418463151nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_8684_gauss__sum,axiom,
    ! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_8685_gauss__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_8686_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(semiri8010041392384452111omplex),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8687_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8688_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one))),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N)),one_on7984719198319812577d_enat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8689_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N))) = times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8690_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8691_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8692_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(semiri5074537144036343181t_real),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8693_sum__gp__offset,axiom,
    ! [X: rat,M: nat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat) ) )
      & ( ( X != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),M)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)) ) ) ) ).

% sum_gp_offset
tff(fact_8694_sum__gp__offset,axiom,
    ! [X: complex,M: nat,N: nat] :
      ( ( ( X = one_one_complex )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex) ) )
      & ( ( X != one_one_complex )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp_offset
tff(fact_8695_sum__gp__offset,axiom,
    ! [X: real,M: nat,N: nat] :
      ( ( ( X = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real) ) )
      & ( ( X != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = 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,X),M)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ).

% sum_gp_offset
tff(fact_8696_log__base__10__eq2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,X)) ) ) ).

% log_base_10_eq2
tff(fact_8697_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N)) = divide6298287555418463151nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8698_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8699_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8700_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_qn(R2),set_or1269000886237332187st_nat(zero_zero_nat,M)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M))),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_8701_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,M: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_qo(R2)),set_or1269000886237332187st_nat(zero_zero_nat,M)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M))),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_8702_gchoose__row__sum__weighted,axiom,
    ! [R2: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qp,R2)),set_or1269000886237332187st_nat(zero_zero_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_8703_and__int_Opinduct,axiom,
    ! [A0: int,A12: int,P: fun_int_fun_int_bool] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(A0,A12)))
     => ( ! [K2: int,L4: int] :
            ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(K2,L4)))
           => ( ( ~ ( pp(aa_set_int_bool(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                    & pp(aa_set_int_bool(member_int(L4),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L4),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,K2),L4)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A12)) ) ) ).

% and_int.pinduct
tff(fact_8704_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(X),Xa2) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
               => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)) ) )
              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
               => ( Y = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X))) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_8705_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: fun_nat_fun_nat_nat,A12: nat,A23: nat,A32: nat,P: fun_fu7164412255009898145t_bool] :
      ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(A0,produc487386426758144856at_nat(A12,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A23),A32))))
     => ( ! [F3: fun_nat_fun_nat_nat,A: nat,B: nat,Acc: nat] :
            ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(F3,produc487386426758144856at_nat(A,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),Acc))))
           => ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aa_fun2305595801590362634t_bool(P,F3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F3,A),Acc))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aa_fun2305595801590362634t_bool(P,F3),A),B),Acc)) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aa_fun2305595801590362634t_bool(P,A0),A12),A23),A32)) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_8706_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X: fun_Pr2477239776880934807at_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(X,Xa2,Xb) = Y )
     => ( accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(X,produc488173922507101015at_nat(Xa2,Xb)))
       => ( ( ( Xa2 = none_P5556105721700978146at_nat )
           => ( ( Y = none_P5556105721700978146at_nat )
             => ~ accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(X,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Xb))) ) )
         => ( ! [V2: product_prod_nat_nat] :
                ( ( Xa2 = some_P7363390416028606310at_nat(V2) )
               => ( ( Xb = none_P5556105721700978146at_nat )
                 => ( ( Y = none_P5556105721700978146at_nat )
                   => ~ accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(X,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))) ) ) )
           => ~ ! [A: product_prod_nat_nat] :
                  ( ( Xa2 = some_P7363390416028606310at_nat(A) )
                 => ! [B: product_prod_nat_nat] :
                      ( ( Xb = some_P7363390416028606310at_nat(B) )
                     => ( ( Y = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(X,A),B)) )
                       => ~ accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(X,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A),some_P7363390416028606310at_nat(B)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_8707_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X: fun_num_fun_num_num,Xa2: option_num,Xb: option_num,Y: option_num] :
      ( ( vEBT_V819420779217536731ft_num(X,Xa2,Xb) = Y )
     => ( accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(X,produc8585076106096196333on_num(Xa2,Xb)))
       => ( ( ( Xa2 = none_num )
           => ( ( Y = none_num )
             => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(X,produc8585076106096196333on_num(none_num,Xb))) ) )
         => ( ! [V2: num] :
                ( ( Xa2 = some_num(V2) )
               => ( ( Xb = none_num )
                 => ( ( Y = none_num )
                   => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(X,produc8585076106096196333on_num(some_num(V2),none_num))) ) ) )
           => ~ ! [A: num] :
                  ( ( Xa2 = some_num(A) )
                 => ! [B: num] :
                      ( ( Xb = some_num(B) )
                     => ( ( Y = some_num(aa_num_num(aa_num_fun_num_num(X,A),B)) )
                       => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(X,produc8585076106096196333on_num(some_num(A),some_num(B)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_8708_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa2: option_nat,Xb: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(X),Xa2),Xb) = Y )
     => ( accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(X,produc5098337634421038937on_nat(Xa2,Xb)))
       => ( ( ( Xa2 = none_nat )
           => ( ( Y = none_nat )
             => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(X,produc5098337634421038937on_nat(none_nat,Xb))) ) )
         => ( ! [V2: nat] :
                ( ( Xa2 = some_nat(V2) )
               => ( ( Xb = none_nat )
                 => ( ( Y = none_nat )
                   => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(X,produc5098337634421038937on_nat(some_nat(V2),none_nat))) ) ) )
           => ~ ! [A: nat] :
                  ( ( Xa2 = some_nat(A) )
                 => ! [B: nat] :
                      ( ( Xb = some_nat(B) )
                     => ( ( Y = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(X,A),B)) )
                       => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(X,produc5098337634421038937on_nat(some_nat(A),some_nat(B)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_8709_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_8710_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_8711_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_8712_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_8713_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_8714_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_8715_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5052692396658037445od_int(M,one) = product_Pair_int_int(aa_num_int(numeral_numeral_int,M),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_8716_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique3479559517661332726nteger(M,one) = produc1086072967326762835nteger(numera6620942414471956472nteger(M),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_8717_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5055182867167087721od_nat(M,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,M)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_8718_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_8719_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(3)
tff(fact_8720_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_8721_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_8722_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(4)
tff(fact_8723_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_8724_sum__multicount__gen,axiom,
    ! [S2: set_bool,T: set_bool,R: fun_bo1841576699660343469l_bool,K: fun_bool_nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S2))
     => ( pp(aa_set_bool_bool(finite_finite_bool,T))
       => ( ! [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),T))
             => ( finite_card_bool(collect_bool(aa_boo8970175072276269406l_bool(aa_fun1447330757145981436l_bool(aTP_Lamm_qq(S2),R),X5))) = aa_bool_nat(K,X5) ) )
         => ( groups5768501273728744877ol_nat(aa_fun3941158123937086921ol_nat(aTP_Lamm_qs(T),R),S2) = groups5768501273728744877ol_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_8725_sum__multicount__gen,axiom,
    ! [S2: set_bool,T: set_real,R: fun_bo2025495424286513072l_bool,K: fun_real_nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S2))
     => ( pp(aa_set_real_bool(finite_finite_real,T))
       => ( ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),T))
             => ( finite_card_bool(collect_bool(aa_rea6668032140610767853l_bool(aa_fun2865020728784200730l_bool(aTP_Lamm_qt(S2),R),X5))) = aa_real_nat(K,X5) ) )
         => ( groups5768501273728744877ol_nat(aa_fun2932017502249704332ol_nat(aTP_Lamm_qv(T),R),S2) = groups1935376822645274424al_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_8726_sum__multicount__gen,axiom,
    ! [S2: set_real,T: set_bool,R: fun_re3316061561565328172l_bool,K: fun_bool_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_bool_bool(finite_finite_bool,T))
       => ( ! [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),T))
             => ( finite_card_real(collect_real(aa_boo2589871674041770537l_bool(aa_fun485223091692333082l_bool(aTP_Lamm_qw(S2),R),X5))) = aa_bool_nat(K,X5) ) )
         => ( groups1935376822645274424al_nat(aa_fun4868894756110751843al_nat(aTP_Lamm_qy(T),R),S2) = groups5768501273728744877ol_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_8727_sum__multicount__gen,axiom,
    ! [S2: set_real,T: set_real,R: fun_re1255401033344146673l_bool,K: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_real_bool(finite_finite_real,T))
       => ( ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),T))
             => ( finite_card_real(collect_real(aa_rea7809428808904475482l_bool(aa_fun1425803159125577980l_bool(aTP_Lamm_qz(S2),R),X5))) = aa_real_nat(K,X5) ) )
         => ( groups1935376822645274424al_nat(aa_fun3074920444450379950al_nat(aTP_Lamm_rb(T),R),S2) = groups1935376822645274424al_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_8728_sum__multicount__gen,axiom,
    ! [S2: set_bool,T: set_int,R: fun_bo2827871560234191408t_bool,K: fun_int_nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S2))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),T))
             => ( finite_card_bool(collect_bool(aa_int_fun_bool_bool(aa_fun6035399485891414554l_bool(aTP_Lamm_df(S2),R),X5))) = aa_int_nat(K,X5) ) )
         => ( groups5768501273728744877ol_nat(aa_fun7155190969624455820ol_nat(aTP_Lamm_rd(T),R),S2) = groups4541462559716669496nt_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_8729_sum__multicount__gen,axiom,
    ! [S2: set_real,T: set_int,R: fun_re8661203372455305201t_bool,K: fun_int_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),T))
             => ( finite_card_real(collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_dg(S2),R),X5))) = aa_int_nat(K,X5) ) )
         => ( groups1935376822645274424al_nat(aa_fun4707068664741690670al_nat(aTP_Lamm_rf(T),R),S2) = groups4541462559716669496nt_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_8730_sum__multicount__gen,axiom,
    ! [S2: set_bool,T: set_complex,R: fun_bo8977660800532043054x_bool,K: fun_complex_nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),T))
             => ( finite_card_bool(collect_bool(aa_com4332222969190720879l_bool(aa_fun6951469949174074266l_bool(aTP_Lamm_dh(S2),R),X5))) = aa_complex_nat(K,X5) ) )
         => ( groups5768501273728744877ol_nat(aa_fun8646024951575708942ol_nat(aTP_Lamm_rg(T),R),S2) = groups5693394587270226106ex_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_8731_sum__multicount__gen,axiom,
    ! [S2: set_real,T: set_complex,R: fun_re3010880487650781423x_bool,K: fun_complex_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),T))
             => ( finite_card_real(collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_di(S2),R),X5))) = aa_complex_nat(K,X5) ) )
         => ( groups1935376822645274424al_nat(aa_fun5167920629197714732al_nat(aTP_Lamm_rh(T),R),S2) = groups5693394587270226106ex_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_8732_sum__multicount__gen,axiom,
    ! [S2: set_bool,T: set_Extended_enat,R: fun_bo6623066019567229854t_bool,K: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),T))
             => ( finite_card_bool(collect_bool(aa_Ext5859091071952444205l_bool(aa_fun4004702039914900634l_bool(aTP_Lamm_dj(S2),R),X5))) = aa_Extended_enat_nat(K,X5) ) )
         => ( groups5768501273728744877ol_nat(aa_fun85076598157486616ol_nat(aTP_Lamm_rj(T),R),S2) = groups2027974829824023292at_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_8733_sum__multicount__gen,axiom,
    ! [S2: set_real,T: set_Extended_enat,R: fun_re7357811994743134493t_bool,K: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T))
       => ( ! [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),T))
             => ( finite_card_real(collect_real(aa_Ext4143593917056713754l_bool(aa_fun3666023268991729404l_bool(aTP_Lamm_dk(S2),R),X5))) = aa_Extended_enat_nat(K,X5) ) )
         => ( groups1935376822645274424al_nat(aa_fun6765506734994765012al_nat(aTP_Lamm_rl(T),R),S2) = groups2027974829824023292at_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_8734_sum__subtractf__nat,axiom,
    ! [A3: set_real,G3: fun_real_nat,F: fun_real_nat] :
      ( ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(G3,X5)),aa_real_nat(F,X5))) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_rm(G3),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A3)),groups1935376822645274424al_nat(G3,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_8735_sum__subtractf__nat,axiom,
    ! [A3: set_bool,G3: fun_bool_nat,F: fun_bool_nat] :
      ( ! [X5: bool] :
          ( pp(aa_set_bool_bool(member_bool(X5),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(G3,X5)),aa_bool_nat(F,X5))) )
     => ( groups5768501273728744877ol_nat(aa_fun9101744387563128124ol_nat(aTP_Lamm_rn(G3),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5768501273728744877ol_nat(F,A3)),groups5768501273728744877ol_nat(G3,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_8736_sum__subtractf__nat,axiom,
    ! [A3: set_set_nat,G3: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X5: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X5),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(G3,X5)),aa_set_nat_nat(F,X5))) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_ro(G3),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A3)),groups8294997508430121362at_nat(G3,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_8737_sum__subtractf__nat,axiom,
    ! [A3: set_int,G3: fun_int_nat,F: fun_int_nat] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(G3,X5)),aa_int_nat(F,X5))) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_rp(G3),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(G3,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_8738_sum__subtractf__nat,axiom,
    ! [A3: set_nat,G3: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(G3,X5)),aa_nat_nat(F,X5))) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rq(G3),F)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(G3),A3)) ) ) ).

% sum_subtractf_nat
tff(fact_8739_sum__eq__Suc0__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( groups4541462559716669496nt_nat(F,A3) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A3))
            & ( aa_int_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),A3))
               => ( ( X3 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8740_sum__eq__Suc0__iff,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( groups5693394587270226106ex_nat(F,A3) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A3))
            & ( aa_complex_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa3),A3))
               => ( ( X3 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8741_sum__eq__Suc0__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( ( aa_set4749074711808745205at_nat(groups977919841031483927at_nat(F),A3) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A3))
            & ( aa_Pro4243173770044397973at_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: product_prod_nat_nat] :
                ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Xa3),A3))
               => ( ( X3 != Xa3 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8742_sum__eq__Suc0__iff,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( groups2027974829824023292at_nat(F,A3) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A3))
            & ( aa_Extended_enat_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa3),A3))
               => ( ( X3 != Xa3 )
                 => ( aa_Extended_enat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8743_sum__eq__Suc0__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A3))
            & ( aa_nat_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa3),A3))
               => ( ( X3 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8744_sum__SucD,axiom,
    ! [F: fun_nat_nat,A3: set_nat,N: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = aa_nat_nat(suc,N) )
     => ? [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),A3))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X5))) ) ) ).

% sum_SucD
tff(fact_8745_sum__eq__1__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( groups4541462559716669496nt_nat(F,A3) = one_one_nat )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A3))
            & ( aa_int_nat(F,X3) = one_one_nat )
            & ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),A3))
               => ( ( X3 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8746_sum__eq__1__iff,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( groups5693394587270226106ex_nat(F,A3) = one_one_nat )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A3))
            & ( aa_complex_nat(F,X3) = one_one_nat )
            & ! [Xa3: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa3),A3))
               => ( ( X3 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8747_sum__eq__1__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( ( aa_set4749074711808745205at_nat(groups977919841031483927at_nat(F),A3) = one_one_nat )
      <=> ? [X3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A3))
            & ( aa_Pro4243173770044397973at_nat(F,X3) = one_one_nat )
            & ! [Xa3: product_prod_nat_nat] :
                ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Xa3),A3))
               => ( ( X3 != Xa3 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8748_sum__eq__1__iff,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( groups2027974829824023292at_nat(F,A3) = one_one_nat )
      <=> ? [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A3))
            & ( aa_Extended_enat_nat(F,X3) = one_one_nat )
            & ! [Xa3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa3),A3))
               => ( ( X3 != Xa3 )
                 => ( aa_Extended_enat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8749_sum__eq__1__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = one_one_nat )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A3))
            & ( aa_nat_nat(F,X3) = one_one_nat )
            & ! [Xa3: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa3),A3))
               => ( ( X3 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8750_sum__multicount,axiom,
    ! [S: set_bool,T2: set_bool,R: fun_bo1841576699660343469l_bool,K: nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( pp(aa_set_bool_bool(finite_finite_bool,T2))
       => ( ! [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),T2))
             => ( finite_card_bool(collect_bool(aa_boo8970175072276269406l_bool(aa_fun1447330757145981436l_bool(aTP_Lamm_qq(S),R),X5))) = K ) )
         => ( groups5768501273728744877ol_nat(aa_fun3941158123937086921ol_nat(aTP_Lamm_qs(T2),R),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_bool(T2)) ) ) ) ) ).

% sum_multicount
tff(fact_8751_sum__multicount,axiom,
    ! [S: set_bool,T2: set_real,R: fun_bo2025495424286513072l_bool,K: nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( pp(aa_set_real_bool(finite_finite_real,T2))
       => ( ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),T2))
             => ( finite_card_bool(collect_bool(aa_rea6668032140610767853l_bool(aa_fun2865020728784200730l_bool(aTP_Lamm_qt(S),R),X5))) = K ) )
         => ( groups5768501273728744877ol_nat(aa_fun2932017502249704332ol_nat(aTP_Lamm_qv(T2),R),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_real(T2)) ) ) ) ) ).

% sum_multicount
tff(fact_8752_sum__multicount,axiom,
    ! [S: set_real,T2: set_bool,R: fun_re3316061561565328172l_bool,K: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( pp(aa_set_bool_bool(finite_finite_bool,T2))
       => ( ! [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),T2))
             => ( finite_card_real(collect_real(aa_boo2589871674041770537l_bool(aa_fun485223091692333082l_bool(aTP_Lamm_qw(S),R),X5))) = K ) )
         => ( groups1935376822645274424al_nat(aa_fun4868894756110751843al_nat(aTP_Lamm_qy(T2),R),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_bool(T2)) ) ) ) ) ).

% sum_multicount
tff(fact_8753_sum__multicount,axiom,
    ! [S: set_real,T2: set_real,R: fun_re1255401033344146673l_bool,K: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( pp(aa_set_real_bool(finite_finite_real,T2))
       => ( ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),T2))
             => ( finite_card_real(collect_real(aa_rea7809428808904475482l_bool(aa_fun1425803159125577980l_bool(aTP_Lamm_qz(S),R),X5))) = K ) )
         => ( groups1935376822645274424al_nat(aa_fun3074920444450379950al_nat(aTP_Lamm_rb(T2),R),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_real(T2)) ) ) ) ) ).

% sum_multicount
tff(fact_8754_sum__multicount,axiom,
    ! [S: set_bool,T2: set_nat,R: fun_bo3288154511880838028t_bool,K: nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T2))
       => ( ! [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),T2))
             => ( finite_card_bool(collect_bool(aa_nat_fun_bool_bool(aa_fun4163026448321506586l_bool(aTP_Lamm_dd(S),R),X5))) = K ) )
         => ( groups5768501273728744877ol_nat(aa_fun7928520491197679664ol_nat(aTP_Lamm_rr(T2),R),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_set_nat_nat(finite_card_nat,T2)) ) ) ) ) ).

% sum_multicount
tff(fact_8755_sum__multicount,axiom,
    ! [S: set_real,T2: set_nat,R: fun_re9121486324101951821t_bool,K: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T2))
       => ( ! [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),T2))
             => ( finite_card_real(collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_de(S),R),X5))) = K ) )
         => ( groups1935376822645274424al_nat(aa_fun5804151645385689226al_nat(aTP_Lamm_rs(T2),R),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_set_nat_nat(finite_card_nat,T2)) ) ) ) ) ).

% sum_multicount
tff(fact_8756_sum__multicount,axiom,
    ! [S: set_bool,T2: set_int,R: fun_bo2827871560234191408t_bool,K: nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T2))
       => ( ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),T2))
             => ( finite_card_bool(collect_bool(aa_int_fun_bool_bool(aa_fun6035399485891414554l_bool(aTP_Lamm_df(S),R),X5))) = K ) )
         => ( groups5768501273728744877ol_nat(aa_fun7155190969624455820ol_nat(aTP_Lamm_rd(T2),R),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_int(T2)) ) ) ) ) ).

% sum_multicount
tff(fact_8757_sum__multicount,axiom,
    ! [S: set_real,T2: set_int,R: fun_re8661203372455305201t_bool,K: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T2))
       => ( ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),T2))
             => ( finite_card_real(collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_dg(S),R),X5))) = K ) )
         => ( groups1935376822645274424al_nat(aa_fun4707068664741690670al_nat(aTP_Lamm_rf(T2),R),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_int(T2)) ) ) ) ) ).

% sum_multicount
tff(fact_8758_sum__multicount,axiom,
    ! [S: set_bool,T2: set_complex,R: fun_bo8977660800532043054x_bool,K: nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),T2))
             => ( finite_card_bool(collect_bool(aa_com4332222969190720879l_bool(aa_fun6951469949174074266l_bool(aTP_Lamm_dh(S),R),X5))) = K ) )
         => ( groups5768501273728744877ol_nat(aa_fun8646024951575708942ol_nat(aTP_Lamm_rg(T2),R),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_complex(T2)) ) ) ) ) ).

% sum_multicount
tff(fact_8759_sum__multicount,axiom,
    ! [S: set_real,T2: set_complex,R: fun_re3010880487650781423x_bool,K: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
       => ( ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),T2))
             => ( finite_card_real(collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_di(S),R),X5))) = K ) )
         => ( groups1935376822645274424al_nat(aa_fun5167920629197714732al_nat(aTP_Lamm_rh(T2),R),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_complex(T2)) ) ) ) ) ).

% sum_multicount
tff(fact_8760_sum__diff__nat,axiom,
    ! [B2: set_complex,A3: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A3))
       => ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A3)),groups5693394587270226106ex_nat(F,B2)) ) ) ) ).

% sum_diff_nat
tff(fact_8761_sum__diff__nat,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(B2),A3))
       => ( aa_set4749074711808745205at_nat(groups977919841031483927at_nat(F),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set4749074711808745205at_nat(groups977919841031483927at_nat(F),A3)),aa_set4749074711808745205at_nat(groups977919841031483927at_nat(F),B2)) ) ) ) ).

% sum_diff_nat
tff(fact_8762_sum__diff__nat,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),A3))
       => ( groups2027974829824023292at_nat(F,minus_925952699566721837d_enat(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups2027974829824023292at_nat(F,A3)),groups2027974829824023292at_nat(F,B2)) ) ) ) ).

% sum_diff_nat
tff(fact_8763_sum__diff__nat,axiom,
    ! [B2: set_int,A3: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A3))
       => ( groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(F,B2)) ) ) ) ).

% sum_diff_nat
tff(fact_8764_sum__diff__nat,axiom,
    ! [B2: set_nat,A3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A3))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B2)) ) ) ) ).

% sum_diff_nat
tff(fact_8765_sum__diff1__nat,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A3))
       => ( aa_set4749074711808745205at_nat(groups977919841031483927at_nat(F),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set4749074711808745205at_nat(groups977919841031483927at_nat(F),A3)),aa_Pro4243173770044397973at_nat(F,A2)) ) )
      & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A3))
       => ( aa_set4749074711808745205at_nat(groups977919841031483927at_nat(F),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A3),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat))) = aa_set4749074711808745205at_nat(groups977919841031483927at_nat(F),A3) ) ) ) ).

% sum_diff1_nat
tff(fact_8766_sum__diff1__nat,axiom,
    ! [A2: set_nat,A3: set_set_nat,F: fun_set_nat_nat] :
      ( ( pp(aa_set_set_nat_bool(member_set_nat(A2),A3))
       => ( groups8294997508430121362at_nat(F,minus_2163939370556025621et_nat(A3,insert_set_nat(A2,bot_bot_set_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A3)),aa_set_nat_nat(F,A2)) ) )
      & ( ~ pp(aa_set_set_nat_bool(member_set_nat(A2),A3))
       => ( groups8294997508430121362at_nat(F,minus_2163939370556025621et_nat(A3,insert_set_nat(A2,bot_bot_set_set_nat))) = groups8294997508430121362at_nat(F,A3) ) ) ) ).

% sum_diff1_nat
tff(fact_8767_sum__diff1__nat,axiom,
    ! [A2: real,A3: set_real,F: fun_real_nat] :
      ( ( pp(aa_set_real_bool(member_real(A2),A3))
       => ( groups1935376822645274424al_nat(F,minus_minus_set_real(A3,insert_real(A2,bot_bot_set_real))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A3)),aa_real_nat(F,A2)) ) )
      & ( ~ pp(aa_set_real_bool(member_real(A2),A3))
       => ( groups1935376822645274424al_nat(F,minus_minus_set_real(A3,insert_real(A2,bot_bot_set_real))) = groups1935376822645274424al_nat(F,A3) ) ) ) ).

% sum_diff1_nat
tff(fact_8768_sum__diff1__nat,axiom,
    ! [A2: bool,A3: set_bool,F: fun_bool_nat] :
      ( ( pp(aa_set_bool_bool(member_bool(A2),A3))
       => ( groups5768501273728744877ol_nat(F,minus_minus_set_bool(A3,insert_bool(A2,bot_bot_set_bool))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5768501273728744877ol_nat(F,A3)),aa_bool_nat(F,A2)) ) )
      & ( ~ pp(aa_set_bool_bool(member_bool(A2),A3))
       => ( groups5768501273728744877ol_nat(F,minus_minus_set_bool(A3,insert_bool(A2,bot_bot_set_bool))) = groups5768501273728744877ol_nat(F,A3) ) ) ) ).

% sum_diff1_nat
tff(fact_8769_sum__diff1__nat,axiom,
    ! [A2: int,A3: set_int,F: fun_int_nat] :
      ( ( pp(aa_set_int_bool(member_int(A2),A3))
       => ( groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(A2,bot_bot_set_int))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A3)),aa_int_nat(F,A2)) ) )
      & ( ~ pp(aa_set_int_bool(member_int(A2),A3))
       => ( groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A3),insert_int(A2,bot_bot_set_int))) = groups4541462559716669496nt_nat(F,A3) ) ) ) ).

% sum_diff1_nat
tff(fact_8770_sum__diff1__nat,axiom,
    ! [A2: nat,A3: set_nat,F: fun_nat_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(A2),A3))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(A2,bot_bot_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_nat_nat(F,A2)) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(A2),A3))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A3),insert_nat(A2,bot_bot_set_nat))) = aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) ) ) ) ).

% sum_diff1_nat
tff(fact_8771_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( groups7754918857620584856omplex(aTP_Lamm_rt,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_ax(N),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_8772_sum__Un__nat,axiom,
    ! [A3: set_int,B2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(F,B2))),groups4541462559716669496nt_nat(F,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B2))) ) ) ) ).

% sum_Un_nat
tff(fact_8773_sum__Un__nat,axiom,
    ! [A3: set_complex,B2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups5693394587270226106ex_nat(F,sup_sup_set_complex(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(F,A3)),groups5693394587270226106ex_nat(F,B2))),groups5693394587270226106ex_nat(F,inf_inf_set_complex(A3,B2))) ) ) ) ).

% sum_Un_nat
tff(fact_8774_sum__Un__nat,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
       => ( groups2027974829824023292at_nat(F,sup_su4489774667511045786d_enat(A3,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2027974829824023292at_nat(F,A3)),groups2027974829824023292at_nat(F,B2))),groups2027974829824023292at_nat(F,inf_in8357106775501769908d_enat(A3,B2))) ) ) ) ).

% sum_Un_nat
tff(fact_8775_sum__Un__nat,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A3))
     => ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
       => ( aa_set4749074711808745205at_nat(groups977919841031483927at_nat(F),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set4749074711808745205at_nat(groups977919841031483927at_nat(F),A3)),aa_set4749074711808745205at_nat(groups977919841031483927at_nat(F),B2))),aa_set4749074711808745205at_nat(groups977919841031483927at_nat(F),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B2))) ) ) ) ).

% sum_Un_nat
tff(fact_8776_sum__Un__nat,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,F: fun_Pr2871540613828922186at_nat] :
      ( finite4392333629123659920at_nat(A3)
     => ( finite4392333629123659920at_nat(B2)
       => ( groups1900718384385340925at_nat(F,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups1900718384385340925at_nat(F,A3)),groups1900718384385340925at_nat(F,B2))),groups1900718384385340925at_nat(F,aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(inf_in4302113700860409141at_nat,A3),B2))) ) ) ) ).

% sum_Un_nat
tff(fact_8777_sum__Un__nat,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,F: fun_Pr1253225528153422666at_nat] :
      ( finite4343798906461161616at_nat(A3)
     => ( finite4343798906461161616at_nat(B2)
       => ( aa_set168828743186997083at_nat(groups3860910324918113789at_nat(F),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set168828743186997083at_nat(groups3860910324918113789at_nat(F),A3)),aa_set168828743186997083at_nat(groups3860910324918113789at_nat(F),B2))),aa_set168828743186997083at_nat(groups3860910324918113789at_nat(F),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A3),B2))) ) ) ) ).

% sum_Un_nat
tff(fact_8778_sum__Un__nat,axiom,
    ! [A3: set_nat,B2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A3),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B2))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B2))) ) ) ) ).

% sum_Un_nat
tff(fact_8779_sum__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( groups7754918857620584856omplex(aTP_Lamm_rt,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_bs,N))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_8780_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,M)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(M,N) = unique5026877609467782581ep_nat(N,unique5055182867167087721od_nat(M,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_8781_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(M,N) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,M)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(M,N) = unique5024387138958732305ep_int(N,unique5052692396658037445od_int(M,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_8782_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(M)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(M,N) = unique4921790084139445826nteger(N,unique3479559517661332726nteger(M,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_8783_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),N))
     => ( groups4538972089207619220nt_int(aTP_Lamm_ru,set_or1266510415728281911st_int(M,N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_8784_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A2: nat,B3: nat,Acc2: nat] :
      ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(F,produc487386426758144856at_nat(A2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),Acc2))))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A2),B3),Acc2) = Acc2 ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A2),B3),Acc2) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),B3),aa_nat_nat(aa_nat_fun_nat_nat(F,A2),Acc2)) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_8785_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa2: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa2),Xb),Xc) = Y )
     => ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(X,produc487386426758144856at_nat(Xa2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xc))))
       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa2))
               => ( Y = Xc ) )
              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa2))
               => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa2),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa2),Xc)) ) ) )
           => ~ accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(X,produc487386426758144856at_nat(Xa2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_8786_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P: fun_int_fun_int_bool] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(A0,A12)))
     => ( ! [I2: int,J2: int] :
            ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(I2,J2)))
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J2))
               => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J2)) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,I2),J2)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A12)) ) ) ).

% upto.pinduct
tff(fact_8787_lemma__termdiff2,axiom,
    ! [H: rat,Z: rat,N: nat] :
      ( ( H != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),H)),N)),aa_nat_rat(power_power_rat(Z),N))),H)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(power_power_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,H),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rw(H),Z),N),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8788_lemma__termdiff2,axiom,
    ! [H: complex,Z: complex,N: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),N)),aa_nat_complex(power_power_complex(Z),N))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_complex_complex(times_times_complex(H),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ry(H),Z),N)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8789_lemma__termdiff2,axiom,
    ! [H: real,Z: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,H),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_sa(H),Z),N)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8790_diffs__equiv,axiom,
    ! [C: fun_nat_complex,X: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_sb(C),X))
     => pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_sc(C),X)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_sb(C),X)))) ) ).

% diffs_equiv
tff(fact_8791_diffs__equiv,axiom,
    ! [C: fun_nat_real,X: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_sd(C),X))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_se(C),X)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_sd(C),X)))) ) ).

% diffs_equiv
tff(fact_8792_lessThan__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,X) = aa_nat_set_nat(set_ord_lessThan_nat,Y) )
    <=> ( X = Y ) ) ).

% lessThan_eq_iff
tff(fact_8793_lessThan__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( set_ord_lessThan_int(X) = set_ord_lessThan_int(Y) )
    <=> ( X = Y ) ) ).

% lessThan_eq_iff
tff(fact_8794_lessThan__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( set_or5984915006950818249n_real(X) = set_or5984915006950818249n_real(Y) )
    <=> ( X = Y ) ) ).

% lessThan_eq_iff
tff(fact_8795_lessThan__iff,axiom,
    ! [I: bool,K: bool] :
      ( pp(aa_set_bool_bool(member_bool(I),set_or4454636193642143162n_bool(K)))
    <=> pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,I),K)) ) ).

% lessThan_iff
tff(fact_8796_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I),set_or890127255671739683et_nat(K)))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,I),K)) ) ).

% lessThan_iff
tff(fact_8797_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( pp(aa_set_rat_bool(member_rat(I),set_ord_lessThan_rat(K)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),K)) ) ).

% lessThan_iff
tff(fact_8798_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( pp(aa_set_num_bool(member_num(I),set_ord_lessThan_num(K)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,I),K)) ) ).

% lessThan_iff
tff(fact_8799_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_ord_lessThan_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).

% lessThan_iff
tff(fact_8800_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( pp(aa_set_int_bool(member_int(I),set_ord_lessThan_int(K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) ) ).

% lessThan_iff
tff(fact_8801_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( pp(aa_set_real_bool(member_real(I),set_or5984915006950818249n_real(K)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),K)) ) ).

% lessThan_iff
tff(fact_8802_finite__lessThan,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_ord_lessThan_nat,K))) ).

% finite_lessThan
tff(fact_8803_card__lessThan,axiom,
    ! [U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_ord_lessThan_nat,U)) = U ).

% card_lessThan
tff(fact_8804_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_ord_lessThan_rat(X)),set_ord_lessThan_rat(Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_8805_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_ord_lessThan_num(X)),set_ord_lessThan_num(Y)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_8806_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,X)),aa_nat_set_nat(set_ord_lessThan_nat,Y)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_8807_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_lessThan_int(X)),set_ord_lessThan_int(Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_8808_lessThan__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or5984915006950818249n_real(X)),set_or5984915006950818249n_real(Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_8809_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_8810_sum_OlessThan__Suc,axiom,
    ! [G3: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G3,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G3,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_rat(G3,N)) ).

% sum.lessThan_Suc
tff(fact_8811_sum_OlessThan__Suc,axiom,
    ! [G3: fun_nat_int,N: nat] : groups3539618377306564664at_int(G3,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G3,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_int(G3,N)) ).

% sum.lessThan_Suc
tff(fact_8812_sum_OlessThan__Suc,axiom,
    ! [G3: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G3),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G3),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_nat(G3,N)) ).

% sum.lessThan_Suc
tff(fact_8813_sum_OlessThan__Suc,axiom,
    ! [G3: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G3),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G3),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_real(G3,N)) ).

% sum.lessThan_Suc
tff(fact_8814_single__Diff__lessThan,axiom,
    ! [K: bool] : minus_minus_set_bool(insert_bool(K,bot_bot_set_bool),set_or4454636193642143162n_bool(K)) = insert_bool(K,bot_bot_set_bool) ).

% single_Diff_lessThan
tff(fact_8815_single__Diff__lessThan,axiom,
    ! [K: nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(K,bot_bot_set_nat)),aa_nat_set_nat(set_ord_lessThan_nat,K)) = insert_nat(K,bot_bot_set_nat) ).

% single_Diff_lessThan
tff(fact_8816_single__Diff__lessThan,axiom,
    ! [K: int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,insert_int(K,bot_bot_set_int)),set_ord_lessThan_int(K)) = insert_int(K,bot_bot_set_int) ).

% single_Diff_lessThan
tff(fact_8817_single__Diff__lessThan,axiom,
    ! [K: real] : minus_minus_set_real(insert_real(K,bot_bot_set_real),set_or5984915006950818249n_real(K)) = insert_real(K,bot_bot_set_real) ).

% single_Diff_lessThan
tff(fact_8818_lessThan__non__empty,axiom,
    ! [X: int] : set_ord_lessThan_int(X) != bot_bot_set_int ).

% lessThan_non_empty
tff(fact_8819_lessThan__non__empty,axiom,
    ! [X: real] : set_or5984915006950818249n_real(X) != bot_bot_set_real ).

% lessThan_non_empty
tff(fact_8820_infinite__Iio,axiom,
    ! [A2: int] : ~ pp(aa_set_int_bool(finite_finite_int,set_ord_lessThan_int(A2))) ).

% infinite_Iio
tff(fact_8821_infinite__Iio,axiom,
    ! [A2: real] : ~ pp(aa_set_real_bool(finite_finite_real,set_or5984915006950818249n_real(A2))) ).

% infinite_Iio
tff(fact_8822_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_sf,U)) ).

% lessThan_def
tff(fact_8823_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_sg,U)) ).

% lessThan_def
tff(fact_8824_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_sh,U)) ).

% lessThan_def
tff(fact_8825_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bk,U)) ).

% lessThan_def
tff(fact_8826_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_si,U)) ).

% lessThan_def
tff(fact_8827_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_sj,U)) ).

% lessThan_def
tff(fact_8828_Iio__eq__empty__iff,axiom,
    ! [N: bool] :
      ( ( set_or4454636193642143162n_bool(N) = bot_bot_set_bool )
    <=> ( pp(N)
      <=> bot_bot_bool ) ) ).

% Iio_eq_empty_iff
tff(fact_8829_Iio__eq__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N) = bot_bot_set_nat )
    <=> ( N = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
tff(fact_8830_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N: rat] :
      ( pp(aa_set_rat_bool(ord_less_set_rat(set_ord_lessThan_rat(M)),set_ord_lessThan_rat(N)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_8831_lessThan__strict__subset__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_set_num_bool(ord_less_set_num(set_ord_lessThan_num(M)),set_ord_lessThan_num(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_8832_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,M)),aa_nat_set_nat(set_ord_lessThan_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_8833_lessThan__strict__subset__iff,axiom,
    ! [M: int,N: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(set_ord_lessThan_int(M)),set_ord_lessThan_int(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_8834_lessThan__strict__subset__iff,axiom,
    ! [M: real,N: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(set_or5984915006950818249n_real(M)),set_or5984915006950818249n_real(N)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_8835_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_8836_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N) = bot_bot_set_nat )
    <=> ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_8837_finite__nat__bounded,axiom,
    ! [S: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ? [K2: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S),aa_nat_set_nat(set_ord_lessThan_nat,K2))) ) ).

% finite_nat_bounded
tff(fact_8838_finite__nat__iff__bounded,axiom,
    ! [S: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
    <=> ? [K3: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S),aa_nat_set_nat(set_ord_lessThan_nat,K3))) ) ).

% finite_nat_iff_bounded
tff(fact_8839_ivl__disj__int__one_I4_J,axiom,
    ! [L: bool,U: bool] : inf_inf_set_bool(set_or4454636193642143162n_bool(L),set_or1454250544011482060t_bool(L,U)) = bot_bot_set_bool ).

% ivl_disj_int_one(4)
tff(fact_8840_ivl__disj__int__one_I4_J,axiom,
    ! [L: nat,U: nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,L)),set_or1269000886237332187st_nat(L,U)) = bot_bot_set_nat ).

% ivl_disj_int_one(4)
tff(fact_8841_ivl__disj__int__one_I4_J,axiom,
    ! [L: int,U: int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,set_ord_lessThan_int(L)),set_or1266510415728281911st_int(L,U)) = bot_bot_set_int ).

% ivl_disj_int_one(4)
tff(fact_8842_ivl__disj__int__one_I4_J,axiom,
    ! [L: real,U: real] : inf_inf_set_real(set_or5984915006950818249n_real(L),set_or1222579329274155063t_real(L,U)) = bot_bot_set_real ).

% ivl_disj_int_one(4)
tff(fact_8843_sum_Onat__diff__reindex,axiom,
    ! [G3: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sk(G3),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(G3),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nat_diff_reindex
tff(fact_8844_sum_Onat__diff__reindex,axiom,
    ! [G3: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_sl(G3),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_real(groups6591440286371151544t_real(G3),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nat_diff_reindex
tff(fact_8845_sum__diff__distrib,axiom,
    ! [Q: fun_int_nat,P: fun_int_nat,N: int] :
      ( ! [X5: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(Q,X5)),aa_int_nat(P,X5)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(P,set_ord_lessThan_int(N))),groups4541462559716669496nt_nat(Q,set_ord_lessThan_int(N))) = groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_rp(Q),P),set_ord_lessThan_int(N)) ) ) ).

% sum_diff_distrib
tff(fact_8846_sum__diff__distrib,axiom,
    ! [Q: fun_real_nat,P: fun_real_nat,N: real] :
      ( ! [X5: real] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(Q,X5)),aa_real_nat(P,X5)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P,set_or5984915006950818249n_real(N))),groups1935376822645274424al_nat(Q,set_or5984915006950818249n_real(N))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_rm(Q),P),set_or5984915006950818249n_real(N)) ) ) ).

% sum_diff_distrib
tff(fact_8847_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P: fun_nat_nat,N: nat] :
      ( ! [X5: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Q,X5)),aa_nat_nat(P,X5)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rq(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ) ).

% sum_diff_distrib
tff(fact_8848_Iio__Int__singleton,axiom,
    ! [X: bool,K: bool] :
      ( ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,X),K))
       => ( inf_inf_set_bool(set_or4454636193642143162n_bool(K),insert_bool(X,bot_bot_set_bool)) = insert_bool(X,bot_bot_set_bool) ) )
      & ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,X),K))
       => ( inf_inf_set_bool(set_or4454636193642143162n_bool(K),insert_bool(X,bot_bot_set_bool)) = bot_bot_set_bool ) ) ) ).

% Iio_Int_singleton
tff(fact_8849_Iio__Int__singleton,axiom,
    ! [X: rat,K: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),K))
       => ( inf_inf_set_rat(set_ord_lessThan_rat(K),insert_rat(X,bot_bot_set_rat)) = insert_rat(X,bot_bot_set_rat) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),K))
       => ( inf_inf_set_rat(set_ord_lessThan_rat(K),insert_rat(X,bot_bot_set_rat)) = bot_bot_set_rat ) ) ) ).

% Iio_Int_singleton
tff(fact_8850_Iio__Int__singleton,axiom,
    ! [X: num,K: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),K))
       => ( inf_inf_set_num(set_ord_lessThan_num(K),insert_num(X,bot_bot_set_num)) = insert_num(X,bot_bot_set_num) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),K))
       => ( inf_inf_set_num(set_ord_lessThan_num(K),insert_num(X,bot_bot_set_num)) = bot_bot_set_num ) ) ) ).

% Iio_Int_singleton
tff(fact_8851_Iio__Int__singleton,axiom,
    ! [X: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),K))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,K)),insert_nat(X,bot_bot_set_nat)) = insert_nat(X,bot_bot_set_nat) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),K))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,K)),insert_nat(X,bot_bot_set_nat)) = bot_bot_set_nat ) ) ) ).

% Iio_Int_singleton
tff(fact_8852_Iio__Int__singleton,axiom,
    ! [X: int,K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),K))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,set_ord_lessThan_int(K)),insert_int(X,bot_bot_set_int)) = insert_int(X,bot_bot_set_int) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),K))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,set_ord_lessThan_int(K)),insert_int(X,bot_bot_set_int)) = bot_bot_set_int ) ) ) ).

% Iio_Int_singleton
tff(fact_8853_Iio__Int__singleton,axiom,
    ! [X: real,K: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),K))
       => ( inf_inf_set_real(set_or5984915006950818249n_real(K),insert_real(X,bot_bot_set_real)) = insert_real(X,bot_bot_set_real) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),K))
       => ( inf_inf_set_real(set_or5984915006950818249n_real(K),insert_real(X,bot_bot_set_real)) = bot_bot_set_real ) ) ) ).

% Iio_Int_singleton
tff(fact_8854_suminf__le__const,axiom,
    ! [F: fun_nat_int,X: int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,suminf_int(F)),X)) ) ) ).

% suminf_le_const
tff(fact_8855_suminf__le__const,axiom,
    ! [F: fun_nat_nat,X: nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,suminf_nat(F)),X)) ) ) ).

% suminf_le_const
tff(fact_8856_suminf__le__const,axiom,
    ! [F: fun_nat_real,X: real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(F)),X)) ) ) ).

% suminf_le_const
tff(fact_8857_sum_OlessThan__Suc__shift,axiom,
    ! [G3: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G3,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G3,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_pd(G3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_8858_sum_OlessThan__Suc__shift,axiom,
    ! [G3: fun_nat_int,N: nat] : groups3539618377306564664at_int(G3,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G3,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_ny(G3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_8859_sum_OlessThan__Suc__shift,axiom,
    ! [G3: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G3),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G3,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_nw(G3)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_8860_sum_OlessThan__Suc__shift,axiom,
    ! [G3: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G3),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G3,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hv(G3)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_8861_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_sm(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,M)) ).

% sum_lessThan_telescope'
tff(fact_8862_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_sn(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,M)) ).

% sum_lessThan_telescope'
tff(fact_8863_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_so(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,M)) ).

% sum_lessThan_telescope'
tff(fact_8864_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_pe(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F,M)),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8865_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_pf(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),aa_nat_int(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8866_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_pg(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8867_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_int,X: int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => summable_int(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8868_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_nat,X: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => summable_nat(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8869_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_real,X: real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => summable_real(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8870_sum_OatLeast1__atMost__eq,axiom,
    ! [G3: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G3),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_nw(G3)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.atLeast1_atMost_eq
tff(fact_8871_sum_OatLeast1__atMost__eq,axiom,
    ! [G3: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G3),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hv(G3)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.atLeast1_atMost_eq
tff(fact_8872_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [B3: nat,A2: nat,F: fun_nat_fun_nat_nat,Acc2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A2),B3),Acc2) = Acc2 ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),A2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A2),B3),Acc2) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),B3),aa_nat_nat(aa_nat_fun_nat_nat(F,A2),Acc2)) ) ) ) ).

% fold_atLeastAtMost_nat.simps
tff(fact_8873_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa2: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa2),Xb),Xc) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa2))
         => ( Y = Xc ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa2))
         => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa2),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa2),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_8874_finite__enumerate__initial__segment,axiom,
    ! [S: set_Extended_enat,N: nat,S2: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),finite121521170596916366d_enat(inf_in8357106775501769908d_enat(S,set_or8419480210114673929d_enat(S2)))))
       => ( infini7641415182203889163d_enat(inf_in8357106775501769908d_enat(S,set_or8419480210114673929d_enat(S2)),N) = infini7641415182203889163d_enat(S,N) ) ) ) ).

% finite_enumerate_initial_segment
tff(fact_8875_finite__enumerate__initial__segment,axiom,
    ! [S: set_nat,N: nat,S2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),aa_nat_set_nat(set_ord_lessThan_nat,S2)))))
       => ( aa_nat_nat(infini8530281810654367211te_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),aa_nat_set_nat(set_ord_lessThan_nat,S2))),N) = aa_nat_nat(infini8530281810654367211te_nat(S),N) ) ) ) ).

% finite_enumerate_initial_segment
tff(fact_8876_power__diff__1__eq,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),one_one_complex)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_8877_power__diff__1__eq,axiom,
    ! [X: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),N)),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),one_one_rat)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_8878_power__diff__1__eq,axiom,
    ! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),N)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_8879_power__diff__1__eq,axiom,
    ! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_8880_one__diff__power__eq,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_8881_one__diff__power__eq,axiom,
    ! [X: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_8882_one__diff__power__eq,axiom,
    ! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_8883_one__diff__power__eq,axiom,
    ! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_8884_geometric__sum,axiom,
    ! [X: rat,N: nat] :
      ( ( X != one_one_rat )
     => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),N)),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),one_one_rat)) ) ) ).

% geometric_sum
tff(fact_8885_geometric__sum,axiom,
    ! [X: complex,N: nat] :
      ( ( X != one_one_complex )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ) ) ).

% geometric_sum
tff(fact_8886_geometric__sum,axiom,
    ! [X: real,N: nat] :
      ( ( X != one_one_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(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,X),N)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) ) ) ).

% geometric_sum
tff(fact_8887_sum__less__suminf,axiom,
    ! [F: fun_nat_int,N: nat] :
      ( summable_int(F)
     => ( ! [M4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,M4))) )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_int(F))) ) ) ).

% sum_less_suminf
tff(fact_8888_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,N: nat] :
      ( summable_nat(F)
     => ( ! [M4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M4))) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_nat(F))) ) ) ).

% sum_less_suminf
tff(fact_8889_sum__less__suminf,axiom,
    ! [F: fun_nat_real,N: nat] :
      ( summable_real(F)
     => ( ! [M4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M4))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,M4))) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_real(F))) ) ) ).

% sum_less_suminf
tff(fact_8890_sum__gp__strict,axiom,
    ! [X: rat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ) )
      & ( ( X != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),N))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)) ) ) ) ).

% sum_gp_strict
tff(fact_8891_sum__gp__strict,axiom,
    ! [X: complex,N: nat] :
      ( ( ( X = one_one_complex )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_complex(semiri8010041392384452111omplex,N) ) )
      & ( ( X != one_one_complex )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp_strict
tff(fact_8892_sum__gp__strict,axiom,
    ! [X: real,N: nat] :
      ( ( ( X = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ) )
      & ( ( X != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ).

% sum_gp_strict
tff(fact_8893_diff__power__eq__sum,axiom,
    ! [X: complex,N: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))),aa_nat_complex(power_power_complex(Y),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_sp(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_8894_diff__power__eq__sum,axiom,
    ! [X: rat,N: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_sq(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_8895_diff__power__eq__sum,axiom,
    ! [X: int,N: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),aa_nat_nat(suc,N))),aa_nat_int(power_power_int(Y),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_sr(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_8896_diff__power__eq__sum,axiom,
    ! [X: real,N: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ss(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_8897_power__diff__sumr2,axiom,
    ! [X: complex,N: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_st(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_8898_power__diff__sumr2,axiom,
    ! [X: rat,N: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(Y),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_su(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_8899_power__diff__sumr2,axiom,
    ! [X: int,N: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_sv(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_8900_power__diff__sumr2,axiom,
    ! [X: real,N: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_sw(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_8901_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_rat,K4: rat,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,P4)),K4)) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),K4))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),K4))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8902_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_int,K4: int,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,P4)),K4)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K4))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),K4))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8903_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_nat,K4: nat,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,P4)),K4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),K4))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),K4))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8904_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_real,K4: real,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,P4)),K4)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),K4))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),K4))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8905_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,N: nat,I: nat] :
      ( summable_int(F)
     => ( ! [M4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M4))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_int(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8906_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,N: nat,I: nat] :
      ( summable_nat(F)
     => ( ! [M4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M4))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_nat(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8907_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,N: nat,I: nat] :
      ( summable_real(F)
     => ( ! [M4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M4))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M4))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_real(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8908_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A2: nat,B3: nat] : groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(A2,B3)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_sx(F),A2,B3),zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_8909_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A2: nat,B3: nat] : groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(A2,B3)) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_sy(F),A2,B3),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_8910_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A2: nat,B3: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(A2,B3)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_sz(F),A2),B3),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_8911_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A2: nat,B3: nat] : aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(A2,B3)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_ta(F),A2),B3),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_8912_one__diff__power__eq_H,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_tb(X),N)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_8913_one__diff__power__eq_H,axiom,
    ! [X: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_tc(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_8914_one__diff__power__eq_H,axiom,
    ! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_td(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_8915_one__diff__power__eq_H,axiom,
    ! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_te(X),N)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_8916_termdiff__converges,axiom,
    ! [X: real,K4: real,C: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),K4))
     => ( ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X5)),K4))
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_ho(C),X5)) )
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_tf(X),C)) ) ) ).

% termdiff_converges
tff(fact_8917_termdiff__converges,axiom,
    ! [X: complex,K4: real,C: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),K4))
     => ( ! [X5: complex] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X5)),K4))
           => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_hq(C),X5)) )
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_tg(X),C)) ) ) ).

% termdiff_converges
tff(fact_8918_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D6)))),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))),D6)),one_one_nat))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F))) ) ) ).

% sum_pos_lt_pair
tff(fact_8919_in__measure,axiom,
    ! [X: option_nat,Y: option_nat,F: fun_option_nat_nat] :
      ( member4117937158525611210on_nat(produc5098337634421038937on_nat(X,Y),measure_option_nat(F))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_option_nat_nat(F,X)),aa_option_nat_nat(F,Y))) ) ).

% in_measure
tff(fact_8920_in__measure,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,F: fun_se4674745386781188650at_nat] :
      ( member1466754251312161552at_nat(produc9060074326276436823at_nat(X,Y),measur4922264856574889999at_nat(F))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set168828743186997083at_nat(F,X)),aa_set168828743186997083at_nat(F,Y))) ) ).

% in_measure
tff(fact_8921_in__measure,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,F: fun_se4097146774670233028at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),measur1827424007717751593at_nat(F))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set4749074711808745205at_nat(F,X)),aa_set4749074711808745205at_nat(F,Y))) ) ).

% in_measure
tff(fact_8922_in__measure,axiom,
    ! [X: nat,Y: nat,F: fun_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),measure_nat(F)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,X)),aa_nat_nat(F,Y))) ) ).

% in_measure
tff(fact_8923_in__measure,axiom,
    ! [X: int,Y: int,F: fun_int_nat] :
      ( member5262025264175285858nt_int(product_Pair_int_int(X,Y),measure_int(F))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X)),aa_int_nat(F,Y))) ) ).

% in_measure
tff(fact_8924_in__finite__psubset,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( member8277197624267554838et_nat(produc4532415448927165861et_nat(A3,B2),finite_psubset_nat)
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A3),B2))
        & pp(aa_set_nat_bool(finite_finite_nat,B2)) ) ) ).

% in_finite_psubset
tff(fact_8925_in__finite__psubset,axiom,
    ! [A3: set_int,B2: set_int] :
      ( member2572552093476627150et_int(produc6363374080413544029et_int(A3,B2),finite_psubset_int)
    <=> ( pp(aa_set_int_bool(ord_less_set_int(A3),B2))
        & pp(aa_set_int_bool(finite_finite_int,B2)) ) ) ).

% in_finite_psubset
tff(fact_8926_in__finite__psubset,axiom,
    ! [A3: set_complex,B2: set_complex] :
      ( member351165363924911826omplex(produc3790773574474814305omplex(A3,B2),finite8643634255014194347omplex)
    <=> ( pp(aa_set_complex_bool(ord_less_set_complex(A3),B2))
        & pp(aa_set_complex_bool(finite3207457112153483333omplex,B2)) ) ) ).

% in_finite_psubset
tff(fact_8927_in__finite__psubset,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat] :
      ( member4453595087596390480d_enat(produc6639060556116774935d_enat(A3,B2),finite4251489430341359785d_enat)
    <=> ( pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(A3),B2))
        & pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2)) ) ) ).

% in_finite_psubset
tff(fact_8928_in__finite__psubset,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( member1466754251312161552at_nat(produc9060074326276436823at_nat(A3,B2),finite4695646753290404266at_nat)
    <=> ( ord_le2604355607129572851at_nat(A3,B2)
        & finite4343798906461161616at_nat(B2) ) ) ).

% in_finite_psubset
tff(fact_8929_in__finite__psubset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A3,B2),finite469560695537375940at_nat)
    <=> ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A3),B2))
        & pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2)) ) ) ).

% in_finite_psubset
tff(fact_8930_monoI1,axiom,
    ! [X6: fun_nat_real] :
      ( ! [M4: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,M4)),aa_nat_real(X6,N2))) )
     => topolo6980174941875973593q_real(X6) ) ).

% monoI1
tff(fact_8931_monoI1,axiom,
    ! [X6: fun_nat_set_int] :
      ( ! [M4: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(X6,M4)),aa_nat_set_int(X6,N2))) )
     => topolo3100542954746470799et_int(X6) ) ).

% monoI1
tff(fact_8932_monoI1,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [M4: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,M4)),aa_nat_rat(X6,N2))) )
     => topolo4267028734544971653eq_rat(X6) ) ).

% monoI1
tff(fact_8933_monoI1,axiom,
    ! [X6: fun_nat_num] :
      ( ! [M4: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X6,M4)),aa_nat_num(X6,N2))) )
     => topolo1459490580787246023eq_num(X6) ) ).

% monoI1
tff(fact_8934_monoI1,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [M4: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X6,M4)),aa_nat_nat(X6,N2))) )
     => topolo4902158794631467389eq_nat(X6) ) ).

% monoI1
tff(fact_8935_monoI1,axiom,
    ! [X6: fun_nat_int] :
      ( ! [M4: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X6,M4)),aa_nat_int(X6,N2))) )
     => topolo4899668324122417113eq_int(X6) ) ).

% monoI1
tff(fact_8936_monoI2,axiom,
    ! [X6: fun_nat_real] :
      ( ! [M4: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,N2)),aa_nat_real(X6,M4))) )
     => topolo6980174941875973593q_real(X6) ) ).

% monoI2
tff(fact_8937_monoI2,axiom,
    ! [X6: fun_nat_set_int] :
      ( ! [M4: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(X6,N2)),aa_nat_set_int(X6,M4))) )
     => topolo3100542954746470799et_int(X6) ) ).

% monoI2
tff(fact_8938_monoI2,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [M4: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,N2)),aa_nat_rat(X6,M4))) )
     => topolo4267028734544971653eq_rat(X6) ) ).

% monoI2
tff(fact_8939_monoI2,axiom,
    ! [X6: fun_nat_num] :
      ( ! [M4: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X6,N2)),aa_nat_num(X6,M4))) )
     => topolo1459490580787246023eq_num(X6) ) ).

% monoI2
tff(fact_8940_monoI2,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [M4: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X6,N2)),aa_nat_nat(X6,M4))) )
     => topolo4902158794631467389eq_nat(X6) ) ).

% monoI2
tff(fact_8941_monoI2,axiom,
    ! [X6: fun_nat_int] :
      ( ! [M4: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X6,N2)),aa_nat_int(X6,M4))) )
     => topolo4899668324122417113eq_int(X6) ) ).

% monoI2
tff(fact_8942_monoseq__def,axiom,
    ! [X6: fun_nat_real] :
      ( topolo6980174941875973593q_real(X6)
    <=> ( ! [M3: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,M3)),aa_nat_real(X6,N3))) )
        | ! [M3: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,N3)),aa_nat_real(X6,M3))) ) ) ) ).

% monoseq_def
tff(fact_8943_monoseq__def,axiom,
    ! [X6: fun_nat_set_int] :
      ( topolo3100542954746470799et_int(X6)
    <=> ( ! [M3: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N3))
           => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(X6,M3)),aa_nat_set_int(X6,N3))) )
        | ! [M3: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N3))
           => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(X6,N3)),aa_nat_set_int(X6,M3))) ) ) ) ).

% monoseq_def
tff(fact_8944_monoseq__def,axiom,
    ! [X6: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X6)
    <=> ( ! [M3: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,M3)),aa_nat_rat(X6,N3))) )
        | ! [M3: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,N3)),aa_nat_rat(X6,M3))) ) ) ) ).

% monoseq_def
tff(fact_8945_monoseq__def,axiom,
    ! [X6: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X6)
    <=> ( ! [M3: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N3))
           => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X6,M3)),aa_nat_num(X6,N3))) )
        | ! [M3: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N3))
           => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X6,N3)),aa_nat_num(X6,M3))) ) ) ) ).

% monoseq_def
tff(fact_8946_monoseq__def,axiom,
    ! [X6: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X6)
    <=> ( ! [M3: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X6,M3)),aa_nat_nat(X6,N3))) )
        | ! [M3: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X6,N3)),aa_nat_nat(X6,M3))) ) ) ) ).

% monoseq_def
tff(fact_8947_monoseq__def,axiom,
    ! [X6: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X6)
    <=> ( ! [M3: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X6,M3)),aa_nat_int(X6,N3))) )
        | ! [M3: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X6,N3)),aa_nat_int(X6,M3))) ) ) ) ).

% monoseq_def
tff(fact_8948_monoseq__Suc,axiom,
    ! [X6: fun_nat_real] :
      ( topolo6980174941875973593q_real(X6)
    <=> ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,N3)),aa_nat_real(X6,aa_nat_nat(suc,N3))))
        | ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,aa_nat_nat(suc,N3))),aa_nat_real(X6,N3))) ) ) ).

% monoseq_Suc
tff(fact_8949_monoseq__Suc,axiom,
    ! [X6: fun_nat_set_int] :
      ( topolo3100542954746470799et_int(X6)
    <=> ( ! [N3: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(X6,N3)),aa_nat_set_int(X6,aa_nat_nat(suc,N3))))
        | ! [N3: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(X6,aa_nat_nat(suc,N3))),aa_nat_set_int(X6,N3))) ) ) ).

% monoseq_Suc
tff(fact_8950_monoseq__Suc,axiom,
    ! [X6: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X6)
    <=> ( ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,N3)),aa_nat_rat(X6,aa_nat_nat(suc,N3))))
        | ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,aa_nat_nat(suc,N3))),aa_nat_rat(X6,N3))) ) ) ).

% monoseq_Suc
tff(fact_8951_monoseq__Suc,axiom,
    ! [X6: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X6)
    <=> ( ! [N3: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X6,N3)),aa_nat_num(X6,aa_nat_nat(suc,N3))))
        | ! [N3: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X6,aa_nat_nat(suc,N3))),aa_nat_num(X6,N3))) ) ) ).

% monoseq_Suc
tff(fact_8952_monoseq__Suc,axiom,
    ! [X6: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X6)
    <=> ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X6,N3)),aa_nat_nat(X6,aa_nat_nat(suc,N3))))
        | ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X6,aa_nat_nat(suc,N3))),aa_nat_nat(X6,N3))) ) ) ).

% monoseq_Suc
tff(fact_8953_monoseq__Suc,axiom,
    ! [X6: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X6)
    <=> ( ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X6,N3)),aa_nat_int(X6,aa_nat_nat(suc,N3))))
        | ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X6,aa_nat_nat(suc,N3))),aa_nat_int(X6,N3))) ) ) ).

% monoseq_Suc
tff(fact_8954_mono__SucI2,axiom,
    ! [X6: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,aa_nat_nat(suc,N2))),aa_nat_real(X6,N2)))
     => topolo6980174941875973593q_real(X6) ) ).

% mono_SucI2
tff(fact_8955_mono__SucI2,axiom,
    ! [X6: fun_nat_set_int] :
      ( ! [N2: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(X6,aa_nat_nat(suc,N2))),aa_nat_set_int(X6,N2)))
     => topolo3100542954746470799et_int(X6) ) ).

% mono_SucI2
tff(fact_8956_mono__SucI2,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,aa_nat_nat(suc,N2))),aa_nat_rat(X6,N2)))
     => topolo4267028734544971653eq_rat(X6) ) ).

% mono_SucI2
tff(fact_8957_mono__SucI2,axiom,
    ! [X6: fun_nat_num] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X6,aa_nat_nat(suc,N2))),aa_nat_num(X6,N2)))
     => topolo1459490580787246023eq_num(X6) ) ).

% mono_SucI2
tff(fact_8958_mono__SucI2,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X6,aa_nat_nat(suc,N2))),aa_nat_nat(X6,N2)))
     => topolo4902158794631467389eq_nat(X6) ) ).

% mono_SucI2
tff(fact_8959_mono__SucI2,axiom,
    ! [X6: fun_nat_int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X6,aa_nat_nat(suc,N2))),aa_nat_int(X6,N2)))
     => topolo4899668324122417113eq_int(X6) ) ).

% mono_SucI2
tff(fact_8960_mono__SucI1,axiom,
    ! [X6: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,N2)),aa_nat_real(X6,aa_nat_nat(suc,N2))))
     => topolo6980174941875973593q_real(X6) ) ).

% mono_SucI1
tff(fact_8961_mono__SucI1,axiom,
    ! [X6: fun_nat_set_int] :
      ( ! [N2: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(X6,N2)),aa_nat_set_int(X6,aa_nat_nat(suc,N2))))
     => topolo3100542954746470799et_int(X6) ) ).

% mono_SucI1
tff(fact_8962_mono__SucI1,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,N2)),aa_nat_rat(X6,aa_nat_nat(suc,N2))))
     => topolo4267028734544971653eq_rat(X6) ) ).

% mono_SucI1
tff(fact_8963_mono__SucI1,axiom,
    ! [X6: fun_nat_num] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X6,N2)),aa_nat_num(X6,aa_nat_nat(suc,N2))))
     => topolo1459490580787246023eq_num(X6) ) ).

% mono_SucI1
tff(fact_8964_mono__SucI1,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X6,N2)),aa_nat_nat(X6,aa_nat_nat(suc,N2))))
     => topolo4902158794631467389eq_nat(X6) ) ).

% mono_SucI1
tff(fact_8965_mono__SucI1,axiom,
    ! [X6: fun_nat_int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X6,N2)),aa_nat_int(X6,aa_nat_nat(suc,N2))))
     => topolo4899668324122417113eq_int(X6) ) ).

% mono_SucI1
tff(fact_8966_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T6)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(exp_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_th,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_8967_floor__log__nat__eq__powr__iff,axiom,
    ! [B3: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B3)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,N) )
        <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),N)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_8968_of__nat__code,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,N) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_ti,N),zero_zero_rat) ).

% of_nat_code
tff(fact_8969_of__nat__code,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_tj),N),zero_zero_nat) ).

% of_nat_code
tff(fact_8970_of__nat__code,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_int_int(semiri8420488043553186161ux_int(aTP_Lamm_tk,N),zero_zero_int) ).

% of_nat_code
tff(fact_8971_of__nat__code,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_tl),N),zero_zero_real) ).

% of_nat_code
tff(fact_8972_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N)))) ) ).

% central_binomial_lower_bound
tff(fact_8973_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_8974_binomial__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,zero_zero_nat)) = N ).

% binomial_1
tff(fact_8975_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% binomial_eq_0_iff
tff(fact_8976_binomial__n__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_8977_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).

% zero_less_binomial_iff
tff(fact_8978_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).

% fact_mono_nat
tff(fact_8979_fact__ge__self,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_self
tff(fact_8980_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_8981_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_8982_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_8983_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_8984_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% binomial_symmetric
tff(fact_8985_binomial__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R2))) ) ).

% binomial_le_pow
tff(fact_8986_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ).

% zero_less_binomial
tff(fact_8987_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_Suc_0_nat
tff(fact_8988_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),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,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_8989_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),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)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% fact_diff_Suc
tff(fact_8990_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R2))) ) ).

% fact_div_fact_le_pow
tff(fact_8991_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real))) ).

% real_of_int_floor_add_one_gt
tff(fact_8992_floor__eq,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X) = N ) ) ) ).

% floor_eq
tff(fact_8993_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real))) ).

% real_of_int_floor_add_one_ge
tff(fact_8994_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2)))) ).

% real_of_int_floor_gt_diff_one
tff(fact_8995_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2)))) ).

% real_of_int_floor_ge_diff_one
tff(fact_8996_binomial__code,axiom,
    ! [N: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),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,N),K)),one_one_nat)),N),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K)) ) ) ) ) ) ).

% binomial_code
tff(fact_8997_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N))) ).

% binomial_maximum'
tff(fact_8998_binomial__mono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K5)),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K5))) ) ) ).

% binomial_mono
tff(fact_8999_binomial__maximum,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% binomial_maximum
tff(fact_9000_binomial__antimono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K5),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K5)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).

% binomial_antimono
tff(fact_9001_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% binomial_le_pow2
tff(fact_9002_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),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,N),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,N),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_9003_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_9004_floor__eq2,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X) = N ) ) ) ).

% floor_eq2
tff(fact_9005_floor__divide__real__eq__div,axiom,
    ! [B3: int,A2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B3))
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_int_real(ring_1_of_int_real,B3))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_real_int(archim6058952711729229775r_real,A2)),B3) ) ) ).

% floor_divide_real_eq_div
tff(fact_9006_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K)))) ) ).

% binomial_less_binomial_Suc
tff(fact_9007_binomial__strict__antimono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K5),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K5)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).

% binomial_strict_antimono
tff(fact_9008_binomial__strict__mono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K5)),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K5))) ) ) ).

% binomial_strict_mono
tff(fact_9009_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,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,N),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,N),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_9010_square__fact__le__2__fact,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% square_fact_le_2_fact
tff(fact_9011_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ? [B7: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_tm(H),J)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B7),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N)),aa_nat_real(semiri2265585572941072030t_real,N)))) ) ).

% Maclaurin_lemma
tff(fact_9012_floor__log__eq__powr__iff,axiom,
    ! [X: real,B3: real,K: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B3),X)) = K )
        <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B3),aa_int_real(ring_1_of_int_real,K))),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B3),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_9013_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
      & ( aa_real_real(exp_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_th,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ).

% Maclaurin_exp_le
tff(fact_9014_floor__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_9015_floor__log__nat__eq__if,axiom,
    ! [B3: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3))
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B3)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_9016_powr__int,axiom,
    ! [X: real,I: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,I)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,I)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,I)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I)))) ) ) ) ) ).

% powr_int
tff(fact_9017_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),one_one_nat))
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_9018_dvd__1__left,axiom,
    ! [K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K)) ).

% dvd_1_left
tff(fact_9019_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( M = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_9020_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( ( K = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_9021_finite__atMost,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_ord_atMost_nat,K))) ).

% finite_atMost
tff(fact_9022_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_9023_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_9024_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int)) ) ).

% nat_0_iff
tff(fact_9025_nat__le__0,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_9026_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ) ).

% zless_nat_conj
tff(fact_9027_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_9028_nat__zminus__int,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_9029_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = zero_zero_int ) ) ) ).

% int_nat_eq
tff(fact_9030_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_9031_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% zero_less_nat_eq
tff(fact_9032_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or1266510415728281911st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)),one_one_int)) ).

% card_atLeastAtMost_int
tff(fact_9033_nat__ceiling__le__eq,axiom,
    ! [X: real,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X))),A2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,A2))) ) ).

% nat_ceiling_le_eq
tff(fact_9034_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% odd_Suc_minus_one
tff(fact_9035_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ).

% even_diff_nat
tff(fact_9036_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% one_less_nat_eq
tff(fact_9037_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_9038_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)),aa_int_nat(nat2,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A2)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_9039_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_9040_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)),aa_int_nat(nat2,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A2)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_9041_gcd__nat_Oextremum,axiom,
    ! [A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),zero_zero_nat)) ).

% gcd_nat.extremum
tff(fact_9042_gcd__nat_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A2))
        & ( zero_zero_nat != A2 ) ) ).

% gcd_nat.extremum_strict
tff(fact_9043_gcd__nat_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A2))
    <=> ( A2 = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_9044_gcd__nat_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),zero_zero_nat))
        & ( A2 != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_9045_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A2))
     => ( A2 = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_9046_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% dvd_diff_nat
tff(fact_9047_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(nat2,Z)),M))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,M))) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
tff(fact_9048_even__nat__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ) ).

% even_nat_iff
tff(fact_9049_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_9050_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y))) ) ).

% nat_mono
tff(fact_9051_atMost__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = set_or1269000886237332187st_nat(zero_zero_nat,N) ).

% atMost_atLeast0
tff(fact_9052_eq__nat__nat__iff,axiom,
    ! [Z: int,Z6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z6))
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z6) )
        <=> ( Z = Z6 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_9053_all__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ! [X_1: nat] : pp(aa_nat_bool(P,X_1))
    <=> ! [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
         => pp(aa_nat_bool(P,aa_int_nat(nat2,X3))) ) ) ).

% all_nat
tff(fact_9054_ex__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_1: nat] : pp(aa_nat_bool(P,X_1))
    <=> ? [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
          & pp(aa_nat_bool(P,aa_int_nat(nat2,X3))) ) ) ).

% ex_nat
tff(fact_9055_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_9056_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M)) ) ) ).

% dvd_pos_nat
tff(fact_9057_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ) ).

% nat_dvd_not_less
tff(fact_9058_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% dvd_minus_self
tff(fact_9059_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_9060_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M))
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_9061_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% less_eq_dvd_minus
tff(fact_9062_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N)) ) ) ) ).

% dvd_diffD1
tff(fact_9063_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M)) ) ) ) ).

% dvd_diffD
tff(fact_9064_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N))
       => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M)) ) ) ).

% zdvd_not_zless
tff(fact_9065_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% dvd_fact
tff(fact_9066_finite__nat__iff__bounded__le,axiom,
    ! [S: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
    <=> ? [K3: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S),aa_nat_set_nat(set_ord_atMost_nat,K3))) ) ).

% finite_nat_iff_bounded_le
tff(fact_9067_nat__mono__iff,axiom,
    ! [Z: int,W2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ) ).

% nat_mono_iff
tff(fact_9068_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z)) ) ).

% zless_nat_eq_int_zless
tff(fact_9069_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),N))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% nat_le_iff
tff(fact_9070_nat__0__le,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_9071_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z )
    <=> ( ( M = aa_int_nat(nat2,Z) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ) ).

% int_eq_iff
tff(fact_9072_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ) ).

% dvd_imp_le
tff(fact_9073_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_9074_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% dvd_mult_cancel
tff(fact_9075_bezout__add__strong__nat,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 != zero_zero_nat )
     => ? [D6: nat,X5: nat,Y3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D6),A2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D6),B3))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),Y3)),D6) ) ) ) ).

% bezout_add_strong_nat
tff(fact_9076_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),N)) ) ) ).

% zdvd_imp_le
tff(fact_9077_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_9078_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),I))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I))) ) ) ).

% dvd_imp_le_int
tff(fact_9079_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q3) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_9080_real__nat__ceiling__ge,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X))))) ).

% real_nat_ceiling_ge
tff(fact_9081_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_tn,M)))) ) ).

% finite_divisors_nat
tff(fact_9082_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ) ).

% nat_less_eq_zless
tff(fact_9083_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),W2))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z)) ) ) ).

% nat_le_eq_zle
tff(fact_9084_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M = aa_int_nat(nat2,W2) )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( W2 = aa_nat_int(semiri1314217659103216013at_int,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
tff(fact_9085_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( aa_int_nat(nat2,W2) = M )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( W2 = aa_nat_int(semiri1314217659103216013at_int,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
tff(fact_9086_split__nat,axiom,
    ! [P: fun_nat_bool,I: int] :
      ( pp(aa_nat_bool(P,aa_int_nat(nat2,I)))
    <=> ( ! [N3: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => pp(aa_nat_bool(P,N3)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
         => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_nat
tff(fact_9087_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_int_nat(nat2,K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ) ).

% le_nat_iff
tff(fact_9088_nat__add__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z6))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) ) ) ) ).

% nat_add_distrib
tff(fact_9089_nat__mult__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) ) ) ).

% nat_mult_distrib
tff(fact_9090_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_9091_nat__diff__distrib,axiom,
    ! [Z6: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z6))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z6),Z))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) ) ) ) ).

% nat_diff_distrib
tff(fact_9092_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ).

% nat_abs_triangle_ineq
tff(fact_9093_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_9094_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_9095_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),N) ) ) ).

% nat_power_eq
tff(fact_9096_nat__floor__neg,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_9097_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_9098_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N ) ) ) ).

% floor_eq3
tff(fact_9099_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),M))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_9100_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),M))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_9101_le__nat__floor,axiom,
    ! [X: nat,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A2)))) ) ).

% le_nat_floor
tff(fact_9102_dvd__minus__add,axiom,
    ! [Q3: nat,N: nat,R2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q3),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),Q3)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M)),Q3)))) ) ) ) ).

% dvd_minus_add
tff(fact_9103_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),I))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_dvd_imp_le
tff(fact_9104_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,R2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),M))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),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),N) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_9105_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( ( L = zero_zero_int )
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L)) ) ) ).

% mod_int_pos_iff
tff(fact_9106_nat__2,axiom,
    aa_int_nat(nat2,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% nat_2
tff(fact_9107_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_9108_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),M))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),aa_nat_int(semiri1314217659103216013at_int,M))) ) ) ).

% nat_less_iff
tff(fact_9109_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z6))) ) ) ).

% nat_mult_distrib_neg
tff(fact_9110_odd__pos,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% odd_pos
tff(fact_9111_nat__abs__int__diff,axiom,
    ! [A2: nat,B3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
       => ( aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B3)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B3),A2) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
       => ( aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B3)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B3) ) ) ) ).

% nat_abs_int_diff
tff(fact_9112_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N ) ) ) ).

% floor_eq4
tff(fact_9113_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% dvd_power_iff_le
tff(fact_9114_diff__nat__eq__if,axiom,
    ! [Z6: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z6),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) = aa_int_nat(nat2,Z) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z6),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z6)),zero_zero_int)),zero_zero_nat),aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z6))) ) ) ) ).

% diff_nat_eq_if
tff(fact_9115_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_to(M),N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_9116_even__set__encode__iff,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A3)))
      <=> ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) ) ) ).

% even_set_encode_iff
tff(fact_9117_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_atMost_nat,N)),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_9118_polynomial__product__nat,axiom,
    ! [M: nat,A2: fun_nat_nat,N: nat,B3: fun_nat_nat,X: nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_nat(A2,I2) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_nat(B3,J2) = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tp(A2),X)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tp(B3),X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_tr(A2),B3),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_9119_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% even_mod_4_div_2
tff(fact_9120_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = produc2626176000494625587at_nat(aTP_Lamm_ts(L),Qr) ).

% divmod_step_nat_def
tff(fact_9121_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_tt(L),Qr) ).

% divmod_step_int_def
tff(fact_9122_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% odd_mod_4_div_2
tff(fact_9123_Bernoulli__inequality__even,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),N))) ) ).

% Bernoulli_inequality_even
tff(fact_9124_sin__coeff__def,axiom,
    ! [X4: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X4))
       => ( aa_nat_real(sin_coeff,X4) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X4))
       => ( aa_nat_real(sin_coeff,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,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X4),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X4)) ) ) ) ).

% sin_coeff_def
tff(fact_9125_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
         => ~ ! [Va2: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
                     => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
                     => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_9126_divmod__nat__if,axiom,
    ! [N: nat,M: nat] :
      ( ( ( ( N = zero_zero_nat )
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
       => ( divmod_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M) ) )
      & ( ~ ( ( N = zero_zero_nat )
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
       => ( divmod_nat(M,N) = produc2626176000494625587at_nat(aTP_Lamm_tu,divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_9127_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_9128_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M))
       => ( M = N ) ) ) ).

% dvd_antisym
tff(fact_9129_sin__x__le__x,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),X)) ) ).

% sin_x_le_x
tff(fact_9130_sin__le__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),one_one_real)) ).

% sin_le_one
tff(fact_9131_cos__le__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X)),one_one_real)) ).

% cos_le_one
tff(fact_9132_abs__sin__x__le__abs__x,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,X))),aa_real_real(abs_abs_real,X))) ).

% abs_sin_x_le_abs_x
tff(fact_9133_sin__cos__le1,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))))),one_one_real)) ).

% sin_cos_le1
tff(fact_9134_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(sin_real,X))) ) ).

% sin_x_ge_neg_x
tff(fact_9135_sin__ge__minus__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(sin_real,X))) ).

% sin_ge_minus_one
tff(fact_9136_cos__ge__minus__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(cos_real,X))) ).

% cos_ge_minus_one
tff(fact_9137_abs__sin__le__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,X))),one_one_real)) ).

% abs_sin_le_one
tff(fact_9138_abs__cos__le__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,X))),one_one_real)) ).

% abs_cos_le_one
tff(fact_9139_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_gt_zero_02
tff(fact_9140_cos__two__less__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real)) ).

% cos_two_less_zero
tff(fact_9141_cos__is__zero,axiom,
    ? [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X5))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(cos_real,X5) = zero_zero_real )
      & ! [Y4: real] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(cos_real,Y4) = zero_zero_real ) )
         => ( Y4 = X5 ) ) ) ).

% cos_is_zero
tff(fact_9142_cos__two__le__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real)) ).

% cos_two_le_zero
tff(fact_9143_cos__double__less__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X))),one_one_real)) ) ) ).

% cos_double_less_one
tff(fact_9144_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va))) = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va))) = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% vebt_buildup.simps(3)
tff(fact_9145_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat)) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
                       => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
                       => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_9146_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),X))
            & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tv,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_9147_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [T6: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),X))
          & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tv,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_9148_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
      & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tv,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_9149_pi__gt__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),pi)) ).

% pi_gt_zero
tff(fact_9150_pi__not__less__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),zero_zero_real)) ).

% pi_not_less_zero
tff(fact_9151_sin__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),pi))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_gt_zero
tff(fact_9152_pi__ge__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),pi)) ).

% pi_ge_zero
tff(fact_9153_cos__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),pi))
           => ( ( aa_real_real(cos_real,X) = aa_real_real(cos_real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_9154_cos__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),pi))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_9155_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_9156_sin__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_ge_zero
tff(fact_9157_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% cos_monotone_0_pi
tff(fact_9158_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),pi))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_9159_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),pi))
       => ( ( aa_real_real(sin_real,X) = zero_zero_real )
         => ( X = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_9160_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X))) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_9161_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y3))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),pi))
      & ( aa_real_real(sin_real,Y3) = aa_real_real(sin_real,X) )
      & ( aa_real_real(cos_real,Y3) = aa_real_real(cos_real,X) ) ) ).

% sincos_principal_value
tff(fact_9162_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),pi))
     => ( ( aa_real_real(sin_real,X) = zero_zero_real )
      <=> ( X = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_9163_pi__less__4,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) ).

% pi_less_4
tff(fact_9164_pi__ge__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) ).

% pi_ge_two
tff(fact_9165_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X))) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_9166_cos__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ? [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X5))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),pi))
            & ( aa_real_real(cos_real,X5) = Y )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),pi))
                  & ( aa_real_real(cos_real,Y4) = Y ) )
               => ( Y4 = X5 ) ) ) ) ) ).

% cos_total
tff(fact_9167_pi__half__less__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_less_two
tff(fact_9168_pi__half__le__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_le_two
tff(fact_9169_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_9170_pi__half__gt__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% pi_half_gt_zero
tff(fact_9171_cos__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cos_real,X))) ) ) ).

% cos_gt_zero
tff(fact_9172_sin__gt__zero2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_gt_zero2
tff(fact_9173_sin__lt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).

% sin_lt_zero
tff(fact_9174_pi__half__ge__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% pi_half_ge_zero
tff(fact_9175_m2pi__less__pi,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))),pi)) ).

% m2pi_less_pi
tff(fact_9176_sin__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( ( aa_real_real(sin_real,X) = aa_real_real(sin_real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_9177_sin__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_9178_sin__monotone__2pi__le,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X))) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_9179_arctan__ubound,axiom,
    ! [Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% arctan_ubound
tff(fact_9180_sin__le__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,pi),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).

% sin_le_zero
tff(fact_9181_minus__pi__half__less__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),zero_zero_real)) ).

% minus_pi_half_less_zero
tff(fact_9182_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cos_real,X))) ) ) ).

% cos_gt_zero_pi
tff(fact_9183_sin__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).

% sin_less_zero
tff(fact_9184_cos__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,X))) ) ) ).

% cos_ge_zero
tff(fact_9185_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_9186_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X))) ) ) ) ).

% sin_monotone_2pi
tff(fact_9187_sin__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ? [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X5))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
            & ( aa_real_real(sin_real,X5) = Y )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
                  & ( aa_real_real(sin_real,Y4) = Y ) )
               => ( Y4 = X5 ) ) ) ) ) ).

% sin_total
tff(fact_9188_arctan__lbound,axiom,
    ! [Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))) ).

% arctan_lbound
tff(fact_9189_arctan__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% arctan_bounded
tff(fact_9190_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_9191_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),pi))
            & ( X = aa_real_real(cos_real,T6) )
            & ( Y = aa_real_real(sin_real,T6) ) ) ) ) ).

% sincos_total_pi
tff(fact_9192_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T6: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
              & ( X = aa_real_real(cos_real,T6) )
              & ( Y = aa_real_real(sin_real,T6) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_9193_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T6: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
          & ( X = aa_real_real(cos_real,T6) )
          & ( Y = aa_real_real(sin_real,T6) ) ) ) ).

% sincos_total_2pi_le
tff(fact_9194_cos__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( aa_real_real(cos_real,X) = zero_zero_real )
       => ? [N2: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_9195_sin__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( aa_real_real(sin_real,X) = zero_zero_real )
       => ? [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_9196_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
             => ( ( X = aa_real_real(cos_real,T6) )
               => ( Y != aa_real_real(sin_real,T6) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_9197_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),X))
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tw,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_9198_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),zero_zero_real))
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tw,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_9199_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
      & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tw,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_9200_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
             => ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T6)),aa_real_real(sin_real,T6)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_9201_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_9202_lemma__tan__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X5))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(tan_real,X5))) ) ) ).

% lemma_tan_total
tff(fact_9203_tan__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(tan_real,X))) ) ) ).

% tan_gt_zero
tff(fact_9204_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X5))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,X5) = Y ) ) ).

% lemma_tan_total1
tff(fact_9205_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_9206_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X))) ) ) ) ) ) ).

% tan_monotone'
tff(fact_9207_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X))) ) ) ) ).

% tan_monotone
tff(fact_9208_tan__total,axiom,
    ! [Y: real] :
    ? [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X5))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,X5) = Y )
      & ! [Y4: real] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
            & ( aa_real_real(tan_real,Y4) = Y ) )
         => ( Y4 = X5 ) ) ) ).

% tan_total
tff(fact_9209_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,X))) ) ) ).

% tan_pos_pi2_le
tff(fact_9210_tan__total__pos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ? [X5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X5))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,X5) = Y ) ) ) ).

% tan_total_pos
tff(fact_9211_tan__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,X)),zero_zero_real)) ) ) ).

% tan_less_zero
tff(fact_9212_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_9213_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))) ) ) ) ).

% tan_mono_le
tff(fact_9214_tan__bound__pi2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,X))),one_one_real)) ) ).

% tan_bound_pi2
tff(fact_9215_arctan,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).

% arctan
tff(fact_9216_arctan__tan,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,X)) = X ) ) ) ).

% arctan_tan
tff(fact_9217_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( ( aa_real_real(tan_real,X) = Y )
         => ( aa_real_real(arctan,Y) = X ) ) ) ) ).

% arctan_unique
tff(fact_9218_tan__total__pi4,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ? [Z3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))),Z3))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))))
          & ( aa_real_real(tan_real,Z3) = X ) ) ) ).

% tan_total_pi4
tff(fact_9219_sin__tan,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,X)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_9220_cos__tan,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_9221_Maclaurin__sin__bound,axiom,
    ! [X: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,X)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tv,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,X)),N)))) ).

% Maclaurin_sin_bound
tff(fact_9222_cot__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cot_real,X)),zero_zero_real)) ) ) ).

% cot_less_zero
tff(fact_9223_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% real_sqrt_less_iff
tff(fact_9224_real__sqrt__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% real_sqrt_le_iff
tff(fact_9225_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y)) ) ).

% real_sqrt_gt_0_iff
tff(fact_9226_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_9227_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y)) ) ).

% real_sqrt_ge_0_iff
tff(fact_9228_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% real_sqrt_le_0_iff
tff(fact_9229_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_9230_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y)) ) ).

% real_sqrt_gt_1_iff
tff(fact_9231_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ).

% real_sqrt_le_1_iff
tff(fact_9232_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y)) ) ).

% real_sqrt_ge_1_iff
tff(fact_9233_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X ) ) ).

% real_sqrt_pow2
tff(fact_9234_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% real_sqrt_pow2_iff
tff(fact_9235_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y))) ) ).

% real_sqrt_le_mono
tff(fact_9236_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y))) ) ).

% real_sqrt_less_mono
tff(fact_9237_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,X)),X) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)) ) ) ).

% sqrt_divide_self_eq
tff(fact_9238_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sqrt,X))) ) ).

% real_sqrt_gt_zero
tff(fact_9239_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,X))) ) ).

% real_sqrt_ge_zero
tff(fact_9240_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( aa_real_real(sqrt,X) = zero_zero_real )
       => ( X = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_9241_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(sqrt,X))) ) ).

% real_sqrt_ge_one
tff(fact_9242_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,X) ) ) ).

% real_inv_sqrt_pow2
tff(fact_9243_real__div__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(sqrt,X)) = aa_real_real(sqrt,X) ) ) ).

% real_div_sqrt
tff(fact_9244_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_9245_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))))) ).

% le_real_sqrt_sumsq
tff(fact_9246_inverse__powr,axiom,
    ! [Y: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(inverse_inverse_real,Y)),A2) = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A2)) ) ) ).

% inverse_powr
tff(fact_9247_sqrt2__less__2,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% sqrt2_less_2
tff(fact_9248_forall__pos__mono__1,axiom,
    ! [P: fun_real_bool,E2: real] :
      ( ! [D6: real,E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D6),E))
         => ( pp(aa_real_bool(P,D6))
           => pp(aa_real_bool(P,E)) ) )
     => ( ! [N2: nat] : pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(P,E2)) ) ) ) ).

% forall_pos_mono_1
tff(fact_9249_forall__pos__mono,axiom,
    ! [P: fun_real_bool,E2: real] :
      ( ! [D6: real,E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D6),E))
         => ( pp(aa_real_bool(P,D6))
           => pp(aa_real_bool(P,E)) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero_nat )
           => pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2)))) )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(P,E2)) ) ) ) ).

% forall_pos_mono
tff(fact_9250_real__arch__inverse,axiom,
    ! [E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
    <=> ? [N3: nat] :
          ( ( N3 != zero_zero_nat )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N3))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N3))),E2)) ) ) ).

% real_arch_inverse
tff(fact_9251_ln__inverse,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,X)) ) ) ).

% ln_inverse
tff(fact_9252_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(sqrt,Y))) ) ).

% real_less_rsqrt
tff(fact_9253_real__le__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sqrt,Y))) ) ).

% real_le_rsqrt
tff(fact_9254_sqrt__le__D,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% sqrt_le_D
tff(fact_9255_log__inverse,axiom,
    ! [A2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X)) ) ) ) ) ).

% log_inverse
tff(fact_9256_real__le__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),Y)) ) ) ) ).

% real_le_lsqrt
tff(fact_9257_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( aa_real_real(sqrt,X) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_9258_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),U))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),U)) ) ).

% lemma_real_divide_sqrt_less
tff(fact_9259_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_9260_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_9261_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C: real,B3: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B3),D)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_9262_sqrt__ge__absD,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(sqrt,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)) ) ).

% sqrt_ge_absD
tff(fact_9263_exp__plus__inverse__exp,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X))))) ).

% exp_plus_inverse_exp
tff(fact_9264_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),Y)) ) ) ) ).

% real_less_lsqrt
tff(fact_9265_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_9266_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_ge_abs1
tff(fact_9267_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_ge_abs2
tff(fact_9268_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,Y)))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_9269_ln__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_9270_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(inverse_inverse_real,X)))) ) ).

% plus_inverse_ge_2
tff(fact_9271_arsinh__real__aux,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))))) ).

% arsinh_real_aux
tff(fact_9272_real__sqrt__power__even,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_9273_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_9274_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_9275_powr__half__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,X) ) ) ).

% powr_half_sqrt
tff(fact_9276_real__le__x__sinh,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% real_le_x_sinh
tff(fact_9277_real__le__abs__sinh,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% real_le_abs_sinh
tff(fact_9278_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real)) ).

% cos_x_y_le_one
tff(fact_9279_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U)) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_9280_arcosh__real__def,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => ( aa_real_real(arcosh_real,X) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_9281_powr__real__of__int,axiom,
    ! [X: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,N)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,N)) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,N)))) ) ) ) ) ).

% powr_real_of_int
tff(fact_9282_sinh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(inverse_inverse_real,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_9283_cot__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cot_real,X))) ) ) ).

% cot_gt_zero
tff(fact_9284_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U)) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_9285_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X)))
     => ( aa_real_real(sin_real,X) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_9286_cos__arcsin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X)) = 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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_9287_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
     => ( aa_real_real(sin_real,aa_real_real(arccos,Y)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_9288_sin__arccos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arccos,X)) = 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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_9289_le__arcsin__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(arcsin,X)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,Y)),X)) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_9290_cos__arccos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_9291_sin__arcsin,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_9292_cosh__real__pos,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cosh_real,X))) ).

% cosh_real_pos
tff(fact_9293_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,X)) = X ) ) ).

% arcosh_cosh_real
tff(fact_9294_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_9295_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_9296_cosh__real__nonneg,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,X))) ).

% cosh_real_nonneg
tff(fact_9297_cosh__real__ge__1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,X))) ).

% cosh_real_ge_1
tff(fact_9298_sinh__less__cosh__real,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X))) ).

% sinh_less_cosh_real
tff(fact_9299_sinh__le__cosh__real,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X))) ).

% sinh_le_cosh_real
tff(fact_9300_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y))) ) ) ).

% cosh_real_strict_mono
tff(fact_9301_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_9302_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_9303_arccos__le__arccos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X))) ) ) ) ).

% arccos_le_arccos
tff(fact_9304_arccos__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)) )
     => ( ( aa_real_real(arccos,X) = aa_real_real(arccos,Y) )
      <=> ( X = Y ) ) ) ).

% arccos_eq_iff
tff(fact_9305_arccos__le__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ) ).

% arccos_le_mono
tff(fact_9306_arcsin__minus,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => ( aa_real_real(arcsin,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,X)) ) ) ) ).

% arcsin_minus
tff(fact_9307_arcsin__le__arcsin,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y))) ) ) ) ).

% arcsin_le_arcsin
tff(fact_9308_arcsin__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( ( aa_real_real(arcsin,X) = aa_real_real(arcsin,Y) )
        <=> ( X = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_9309_arcsin__le__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ).

% arcsin_le_mono
tff(fact_9310_arccos__lbound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))) ) ) ).

% arccos_lbound
tff(fact_9311_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X))) ) ) ) ).

% arccos_less_arccos
tff(fact_9312_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ).

% arccos_less_mono
tff(fact_9313_arccos__ubound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi)) ) ) ).

% arccos_ubound
tff(fact_9314_arccos__cos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
       => ( aa_real_real(arccos,aa_real_real(cos_real,X)) = X ) ) ) ).

% arccos_cos
tff(fact_9315_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y))) ) ) ) ).

% arcsin_less_arcsin
tff(fact_9316_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).

% arcsin_less_mono
tff(fact_9317_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_9318_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Theta)),pi))
     => ( aa_real_real(arccos,aa_real_real(cos_real,Theta)) = aa_real_real(abs_abs_real,Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_9319_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y)),pi)) ) ) ) ).

% arccos_lt_bounded
tff(fact_9320_arccos__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi)) ) ) ) ).

% arccos_bounded
tff(fact_9321_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arccos,X)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_9322_arccos__cos2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),X))
       => ( aa_real_real(arccos,aa_real_real(cos_real,X)) = aa_real_real(uminus_uminus_real,X) ) ) ) ).

% arccos_cos2
tff(fact_9323_arccos__minus,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,X)) ) ) ) ).

% arccos_minus
tff(fact_9324_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_9325_arccos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi))
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_9326_arccos__minus__abs,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,X)) ) ) ).

% arccos_minus_abs
tff(fact_9327_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ).

% arccos_le_pi2
tff(fact_9328_cosh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(inverse_inverse_real,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_9329_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_9330_arcsin__lbound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))) ) ) ).

% arcsin_lbound
tff(fact_9331_arcsin__ubound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ).

% arcsin_ubound
tff(fact_9332_arcsin__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arcsin_bounded
tff(fact_9333_arcsin__sin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,X)) = X ) ) ) ).

% arcsin_sin
tff(fact_9334_arcsin,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_9335_arcsin__pi,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),pi))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_9336_arcsin__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X)),Y))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sin_real,Y))) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_9337_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_tx(D))) ).

% int_ge_less_than2_def
tff(fact_9338_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_ty(D))) ).

% int_ge_less_than_def
tff(fact_9339_set__decode__0,axiom,
    ! [X: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),nat_set_decode(X)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)) ) ).

% set_decode_0
tff(fact_9340_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_tz,Y)) ).

% arctan_def
tff(fact_9341_set__decode__zero,axiom,
    nat_set_decode(zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_9342_set__encode__inverse,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( nat_set_decode(aa_set_nat_nat(nat_set_encode,A3)) = A3 ) ) ).

% set_encode_inverse
tff(fact_9343_finite__set__decode,axiom,
    ! [N: nat] : pp(aa_set_nat_bool(finite_finite_nat,nat_set_decode(N))) ).

% finite_set_decode
tff(fact_9344_ln__neg__is__const,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( aa_real_real(ln_ln_real,X) = the_real(aTP_Lamm_er) ) ) ).

% ln_neg_is_const
tff(fact_9345_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,nat_set_decode(M)),nat_set_decode(N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% subset_decode_imp_le
tff(fact_9346_arccos__def,axiom,
    ! [Y: real] : aa_real_real(arccos,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ua,Y)) ).

% arccos_def
tff(fact_9347_pi__half,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = the_real(aTP_Lamm_ub) ).

% pi_half
tff(fact_9348_pi__def,axiom,
    pi = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),the_real(aTP_Lamm_ub)) ).

% pi_def
tff(fact_9349_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_uc,Y)) ).

% arcsin_def
tff(fact_9350_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : aa_num_nat(size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(3)
tff(fact_9351_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M)) ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_9352_num_Osize__gen_I2_J,axiom,
    ! [X2: num] : aa_num_nat(size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_9353_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_Suc_0
tff(fact_9354_Suc__0__mod__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),aa_nat_nat(suc,zero_zero_nat)))) ).

% Suc_0_mod_eq
tff(fact_9355_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),Q3))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_9356_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),M)) ).

% take_bit_nat_less_eq_self
tff(fact_9357_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K)) ) ) ).

% take_bit_nat_eq
tff(fact_9358_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_9359_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(M),K)),aa_int_int(bit_se2923211474154528505it_int(N),K))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_9360_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(N),K))) ).

% take_bit_nonnegative
tff(fact_9361_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_9362_not__take__bit__negative,axiom,
    ! [N: nat,K: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),zero_zero_int)) ).

% not_take_bit_negative
tff(fact_9363_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(bit_se2923211474154528505it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_9364_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_9365_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% take_bit_nat_less_exp
tff(fact_9366_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_9367_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% take_bit_int_less_exp
tff(fact_9368_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_9369_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),M))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M)) ) ).

% take_bit_nat_less_self_iff
tff(fact_9370_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_se2923211474154528505it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_9371_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K)) ) ).

% take_bit_int_less_self_iff
tff(fact_9372_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = K )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_9373_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_9374_floor__real__def,axiom,
    ! [X: real] : aa_real_int(archim6058952711729229775r_real,X) = the_int(aa_real_fun_int_bool(aTP_Lamm_ud,X)) ).

% floor_real_def
tff(fact_9375_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_9376_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_int_int(bit_se2923211474154528505it_int(N),K))) ) ).

% take_bit_int_greater_eq
tff(fact_9377_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,K)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_9378_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_int ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_9379_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2159334234014336723it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_9380_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2159334234014336723it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% flip_bit_negative_int_iff
tff(fact_9381_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(K,L)))
     => ( ( ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))) ) )
        & ( ~ ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
              & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_9382_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa2) = Y )
     => ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(X,Xa2)))
       => ~ ( ( ( ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                  & pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => ( Y = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))) ) )
              & ( ~ ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                    & pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => ( Y = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) )
           => ~ pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(X,Xa2))) ) ) ) ).

% and_int.pelims
tff(fact_9383_floor__rat__def,axiom,
    ! [X: rat] : aa_rat_int(archim3151403230148437115or_rat,X) = the_int(aTP_Lamm_ue(X)) ).

% floor_rat_def
tff(fact_9384_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      ( ( ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
          & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
       => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))) ) )
      & ( ~ ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
       => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.simps
tff(fact_9385_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% and_nonnegative_int_iff
tff(fact_9386_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),zero_zero_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% and_negative_int_iff
tff(fact_9387_abs__rat__def,axiom,
    ! [A2: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A2) = aa_rat_rat(uminus_uminus_rat,A2) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A2) = A2 ) ) ) ).

% abs_rat_def
tff(fact_9388_sgn__rat__def,axiom,
    ! [A2: rat] :
      ( ( ( A2 = zero_zero_rat )
       => ( aa_rat_rat(sgn_sgn_rat,A2) = zero_zero_rat ) )
      & ( ( A2 != zero_zero_rat )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
           => ( aa_rat_rat(sgn_sgn_rat,A2) = one_one_rat ) )
          & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
           => ( aa_rat_rat(sgn_sgn_rat,A2) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ) ) ).

% sgn_rat_def
tff(fact_9389_less__eq__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | ( X = Y ) ) ) ).

% less_eq_rat_def
tff(fact_9390_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
     => ~ ! [S3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),S3))
           => ! [T6: rat] :
                ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),T6))
               => ( R2 != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,S3),T6) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9391_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y))) ) ).

% AND_lower
tff(fact_9392_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),X)) ) ).

% AND_upper1
tff(fact_9393_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),Y)) ) ).

% AND_upper2
tff(fact_9394_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z)) ) ) ).

% AND_upper1'
tff(fact_9395_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),Z)) ) ) ).

% AND_upper2'
tff(fact_9396_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),Z)) ) ) ).

% AND_upper2''
tff(fact_9397_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z)) ) ) ).

% AND_upper1''
tff(fact_9398_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),K)) ) ).

% and_less_eq
tff(fact_9399_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa2) = Y )
     => ( ( ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( Y = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))) ) )
        & ( ~ ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
              & pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( Y = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.elims
tff(fact_9400_normalize__negative,axiom,
    ! [Q3: int,P3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Q3),zero_zero_int))
     => ( normalize(product_Pair_int_int(P3,Q3)) = normalize(product_Pair_int_int(aa_int_int(uminus_uminus_int,P3),aa_int_int(uminus_uminus_int,Q3))) ) ) ).

% normalize_negative
tff(fact_9401_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% xor_Suc_0_eq
tff(fact_9402_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% Suc_0_xor_eq
tff(fact_9403_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_9404_and__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_9405_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_9406_and__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_9407_and__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_9408_Suc__0__and__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_9409_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_9410_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y)) ).

% xor_nat_numerals(2)
tff(fact_9411_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% xor_nat_numerals(3)
tff(fact_9412_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X)) ).

% xor_nat_numerals(4)
tff(fact_9413_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P3: int,Q3: int] :
      ( ( normalize(R2) = product_Pair_int_int(P3,Q3) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q3)) ) ).

% normalize_denom_pos
tff(fact_9414_and__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( M = zero_zero_nat )
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = zero_zero_nat ) )
      & ( ~ ( ( M = zero_zero_nat )
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% and_nat_unfold
tff(fact_9415_xor__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = M ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_9416_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups4048646594741130792ol_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),size_size_list_bool(Bs)))) ).

% horner_sum_of_bool_2_less
tff(fact_9417_Least__eq__0,axiom,
    ! [P: fun_nat_bool] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ord_Least_nat(P) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_9418_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% xor_nonnegative_int_iff
tff(fact_9419_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)),zero_zero_int))
    <=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% xor_negative_int_iff
tff(fact_9420_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),Y))) ) ) ).

% XOR_lower
tff(fact_9421_Least__Suc2,axiom,
    ! [P: fun_nat_bool,N: nat,Q: fun_nat_bool,M: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( pp(aa_nat_bool(Q,M))
       => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
         => ( ! [K2: nat] :
                ( pp(aa_nat_bool(P,aa_nat_nat(suc,K2)))
              <=> pp(aa_nat_bool(Q,K2)) )
           => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_9422_Least__Suc,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_uf(P))) ) ) ) ).

% Least_Suc
tff(fact_9423_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_qb),set_or1269000886237332187st_nat(aa_nat_nat(suc,N),M))) ) ) ).

% fact_eq_fact_times
tff(fact_9424_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_qb),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),M)) ) ) ).

% fact_div_fact
tff(fact_9425_XOR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% XOR_upper
tff(fact_9426_or__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = M ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_9427_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_qb),set_or4665077453230672383an_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_9428_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,set_or4665077453230672383an_nat(L,U))) ).

% finite_atLeastLessThan
tff(fact_9429_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,set_or4665077453230672383an_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_atLeastLessThan
tff(fact_9430_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or4665077453230672383an_nat(M,aa_nat_nat(suc,M)) = insert_nat(M,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_9431_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(4)
tff(fact_9432_or__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(2)
tff(fact_9433_or__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_9434_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(3)
tff(fact_9435_all__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
         => pp(aa_nat_bool(P,M3)) )
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_or4665077453230672383an_nat(zero_zero_nat,N)))
         => pp(aa_nat_bool(P,X3)) ) ) ).

% all_nat_less_eq
tff(fact_9436_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
          & pp(aa_nat_bool(P,M3)) )
    <=> ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_or4665077453230672383an_nat(zero_zero_nat,N)))
          & pp(aa_nat_bool(P,X3)) ) ) ).

% ex_nat_less_eq
tff(fact_9437_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : set_or4665077453230672383an_nat(L,aa_nat_nat(suc,U)) = set_or1269000886237332187st_nat(L,U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_9438_lessThan__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = set_or4665077453230672383an_nat(zero_zero_nat,N) ).

% lessThan_atLeast0
tff(fact_9439_atLeastLessThan0,axiom,
    ! [M: nat] : set_or4665077453230672383an_nat(M,zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_9440_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat(N,set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% atLeast0_lessThan_Suc
tff(fact_9441_prod__Suc__fact,axiom,
    ! [N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(zero_zero_nat,N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_fact
tff(fact_9442_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_Suc_fact
tff(fact_9443_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I,J)) = groups1705073143266064639nt_int(aTP_Lamm_ru,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_9444_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,N4),set_or4665077453230672383an_nat(zero_zero_nat,N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_9445_subset__card__intvl__is__intvl,axiom,
    ! [A3: set_nat,K: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),set_or4665077453230672383an_nat(K,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A3)))))
     => ( A3 = set_or4665077453230672383an_nat(K,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A3))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_9446_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( set_or4665077453230672383an_nat(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,set_or4665077453230672383an_nat(I,J)),set_or4665077453230672383an_nat(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_9447_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = groups1705073143266064639nt_int(aTP_Lamm_ru,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_9448_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N)) = insert_nat(N,set_or4665077453230672383an_nat(M,N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
tff(fact_9449_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,N4),set_or4665077453230672383an_nat(zero_zero_nat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N4)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9450_card__sum__le__nat__sum,axiom,
    ! [S: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_qb),set_or4665077453230672383an_nat(zero_zero_nat,aa_set_nat_nat(finite_card_nat,S)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_qb),S))) ).

% card_sum_le_nat_sum
tff(fact_9451_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),N) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,N)),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9452_Suc__0__or__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% Suc_0_or_eq
tff(fact_9453_or__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% or_Suc_0_eq
tff(fact_9454_sum__power2,axiom,
    ! [K: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),set_or4665077453230672383an_nat(zero_zero_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),one_one_nat) ).

% sum_power2
tff(fact_9455_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A2: fun_nat_nat,B3: fun_nat_nat] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(A2,I2)),aa_nat_nat(A2,J2))) ) )
     => ( ! [I2: nat,J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(B3,J2)),aa_nat_nat(B3,I2))) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ug(A2),B3)),set_or4665077453230672383an_nat(zero_zero_nat,N)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A2),set_or4665077453230672383an_nat(zero_zero_nat,N))),aa_set_nat_nat(groups3542108847815614940at_nat(B3),set_or4665077453230672383an_nat(zero_zero_nat,N))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9456_Cauchy__iff2,axiom,
    ! [X6: fun_nat_real] :
      ( topolo4055970368930404560y_real(X6)
    <=> ! [J3: nat] :
        ? [M8: nat] :
        ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),M3))
         => ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),N3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X6,M3)),aa_nat_real(X6,N3)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3))))) ) ) ) ).

% Cauchy_iff2
tff(fact_9457_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,set_or4662586982721622107an_int(L,U))) ).

% finite_atLeastLessThan_int
tff(fact_9458_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% or_nonnegative_int_iff
tff(fact_9459_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)),zero_zero_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% or_negative_int_iff
tff(fact_9460_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or4662586982721622107an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).

% card_atLeastLessThan_int
tff(fact_9461_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y))) ) ) ).

% OR_lower
tff(fact_9462_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L))) ) ).

% or_greater_eq
tff(fact_9463_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : pp(aa_set_int_bool(finite_finite_int,set_or4662586982721622107an_int(zero_zero_int,U))) ).

% finite_atLeastZeroLessThan_int
tff(fact_9464_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(L,aa_int_int(aa_int_fun_int_int(plus_plus_int,U),one_one_int)) = set_or1266510415728281911st_int(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_9465_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_9466_OR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% OR_upper
tff(fact_9467_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(vEBT_size_VEBT,X13)),aa_VEBT_VEBT_nat(vEBT_size_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size_gen(1)
tff(fact_9468_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(size_size_VEBT_VEBT,X13)),aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size(3)
tff(fact_9469_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(I,J)))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
         => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
         => ( upto(I,J) = nil_int ) ) ) ) ).

% upto.psimps
tff(fact_9470_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( upto(X,Xa2) = Y )
     => ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(X,Xa2)))
       => ~ ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
               => ( Y = cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa2)) ) )
              & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
               => ( Y = nil_int ) ) )
           => ~ pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(X,Xa2))) ) ) ) ).

% upto.pelims
tff(fact_9471_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( 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,N))) = cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_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,N)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( 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,N))) = nil_int ) ) ) ).

% upto_rec_numeral(4)
tff(fact_9472_upto__empty,axiom,
    ! [J: int,I: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
     => ( upto(I,J) = nil_int ) ) ).

% upto_empty
tff(fact_9473_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = upto(I,J) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I)) ) ).

% upto_Nil2
tff(fact_9474_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I)) ) ).

% upto_Nil
tff(fact_9475_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K))),J))
     => ( aa_nat_int(nth_int(upto(I,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_9476_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_num_int(numeral_numeral_int,N))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).

% upto_rec_numeral(1)
tff(fact_9477_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_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,N)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = nil_int ) ) ) ).

% upto_rec_numeral(2)
tff(fact_9478_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_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,N))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).

% upto_rec_numeral(3)
tff(fact_9479_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_9480_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_9481_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9482_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_9483_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9484_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( upto(X,Xa2) = Y )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
         => ( Y = cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa2)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
tff(fact_9485_upto_Osimps,axiom,
    ! [I: int,J: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
       => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
       => ( upto(I,J) = nil_int ) ) ) ).

% upto.simps
tff(fact_9486_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_uh(L),Qr) ).

% divmod_step_integer_def
tff(fact_9487_less__eq__integer__code_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).

% less_eq_integer_code(1)
tff(fact_9488_sgn__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( sgn_sgn_Code_integer(K) = zero_z3403309356797280102nteger ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
           => ( sgn_sgn_Code_integer(K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
           => ( sgn_sgn_Code_integer(K) = one_one_Code_integer ) ) ) ) ) ).

% sgn_integer_code
tff(fact_9489_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_9490_integer__of__int__code,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_Code_integer(code_integer_of_int,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( ( ( K = zero_zero_int )
           => ( aa_int_Code_integer(code_integer_of_int,K) = zero_z3403309356797280102nteger ) )
          & ( ( K != zero_zero_int )
           => ( aa_int_Code_integer(code_integer_of_int,K) = if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_Code_integer)) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_9491_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri631733984087533419it_int(N),K)))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_9492_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),zero_zero_int))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).

% signed_take_bit_negative_iff
tff(fact_9493_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),N))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ) ).

% bit_nat_iff
tff(fact_9494_less__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa2),X)) ) ).

% less_integer.abs_eq
tff(fact_9495_less__integer__code_I1_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).

% less_integer_code(1)
tff(fact_9496_abs__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(K) = K ) ) ) ).

% abs_integer_code
tff(fact_9497_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,N))) ).

% not_bit_Suc_0_Suc
tff(fact_9498_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),N))
    <=> ( N = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_9499_less__eq__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa2),X)) ) ).

% less_eq_integer.abs_eq
tff(fact_9500_not__bit__Suc__0__numeral,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,N))) ).

% not_bit_Suc_0_numeral
tff(fact_9501_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(M),K))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_9502_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M2))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),M2))
              <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N2)) ) )
         => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)))
              <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N2)) ) ) ) ).

% int_bit_bound
tff(fact_9503_binomial__def,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = finite_card_set_nat(collect_set_nat(aa_nat8857836526276733660t_bool(aTP_Lamm_ui(N),K))) ).

% binomial_def
tff(fact_9504_int__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( code_int_of_integer(K) = aa_int_int(uminus_uminus_int,code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( ( ( K = zero_z3403309356797280102nteger )
           => ( code_int_of_integer(K) = zero_zero_int ) )
          & ( ( K != zero_z3403309356797280102nteger )
           => ( code_int_of_integer(K) = produc1553301316500091796er_int(aTP_Lamm_uj,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ) ) ).

% int_of_integer_code
tff(fact_9505_finite__enumerate,axiom,
    ! [S: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ? [R3: fun_nat_nat] :
          ( pp(aa_set_nat_bool(strict1292158309912662752at_nat(R3),aa_nat_set_nat(set_ord_lessThan_nat,aa_set_nat_nat(finite_card_nat,S))))
          & ! [N5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N5),aa_set_nat_nat(finite_card_nat,S)))
             => pp(aa_set_nat_bool(member_nat(aa_nat_nat(R3,N5)),S)) ) ) ) ).

% finite_enumerate
tff(fact_9506_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,set_or1269000886237332187st_nat(I,J)) = set_or1269000886237332187st_nat(aa_nat_nat(suc,I),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_9507_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,set_or4665077453230672383an_nat(I,J)) = set_or4665077453230672383an_nat(aa_nat_nat(suc,I),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_9508_zero__notin__Suc__image,axiom,
    ! [A3: set_nat] : ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),image_nat_nat(suc,A3))) ).

% zero_notin_Suc_image
tff(fact_9509_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Xa2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,code_int_of_integer(X)),code_int_of_integer(Xa2))) ) ).

% less_integer.rep_eq
tff(fact_9510_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),L))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,code_int_of_integer(K)),code_int_of_integer(L))) ) ).

% integer_less_iff
tff(fact_9511_less__eq__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Xa2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,code_int_of_integer(X)),code_int_of_integer(Xa2))) ) ).

% less_eq_integer.rep_eq
tff(fact_9512_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),L))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,code_int_of_integer(K)),code_int_of_integer(L))) ) ).

% integer_less_eq_iff
tff(fact_9513_finite__int__iff__bounded,axiom,
    ! [S: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
    <=> ? [K3: int] : pp(aa_set_int_bool(ord_less_eq_set_int(image_int_int(abs_abs_int,S)),set_ord_lessThan_int(K3))) ) ).

% finite_int_iff_bounded
tff(fact_9514_finite__int__iff__bounded__le,axiom,
    ! [S: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
    <=> ? [K3: int] : pp(aa_set_int_bool(ord_less_eq_set_int(image_int_int(abs_abs_int,S)),set_ord_atMost_int(K3))) ) ).

% finite_int_iff_bounded_le
tff(fact_9515_image__int__atLeastAtMost,axiom,
    ! [A2: nat,B3: nat] : image_nat_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(A2,B3)) = set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,A2),aa_nat_int(semiri1314217659103216013at_int,B3)) ).

% image_int_atLeastAtMost
tff(fact_9516_image__int__atLeastLessThan,axiom,
    ! [A2: nat,B3: nat] : image_nat_int(semiri1314217659103216013at_int,set_or4665077453230672383an_nat(A2,B3)) = set_or4662586982721622107an_int(aa_nat_int(semiri1314217659103216013at_int,A2),aa_nat_int(semiri1314217659103216013at_int,B3)) ).

% image_int_atLeastLessThan
tff(fact_9517_image__Suc__lessThan,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N)) = set_or1269000886237332187st_nat(one_one_nat,N) ).

% image_Suc_lessThan
tff(fact_9518_image__Suc__atMost,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N)) = set_or1269000886237332187st_nat(one_one_nat,aa_nat_nat(suc,N)) ).

% image_Suc_atMost
tff(fact_9519_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or1269000886237332187st_nat(zero_zero_nat,N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9520_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or4665077453230672383an_nat(zero_zero_nat,N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9521_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9522_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% atMost_Suc_eq_insert_0
tff(fact_9523_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_uk,L),set_or4662586982721622107an_int(zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L))) = set_or4662586982721622107an_int(L,U) ).

% image_add_int_atLeastLessThan
tff(fact_9524_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
     => ( set_or4662586982721622107an_int(zero_zero_int,U) = image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_9525_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ul,C),set_or4665077453230672383an_nat(X,Y)) = set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ul,C),set_or4665077453230672383an_nat(X,Y)) = insert_nat(zero_zero_nat,bot_bot_set_nat) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ul,C),set_or4665077453230672383an_nat(X,Y)) = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9526_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_9527_num__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),one_one_Code_integer))
       => ( code_num_of_integer(K) = one ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),one_one_Code_integer))
       => ( code_num_of_integer(K) = produc7336495610019696514er_num(aTP_Lamm_um,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).

% num_of_integer_code
tff(fact_9528_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
       => ( code_nat_of_integer(K) = zero_zero_nat ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
       => ( code_nat_of_integer(K) = produc1555791787009142072er_nat(aTP_Lamm_un,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).

% nat_of_integer_code
tff(fact_9529_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
     => ( code_nat_of_integer(K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_9530_nat__of__integer__code__post_I1_J,axiom,
    code_nat_of_integer(zero_z3403309356797280102nteger) = zero_zero_nat ).

% nat_of_integer_code_post(1)
tff(fact_9531_Inf__nat__def1,axiom,
    ! [K4: set_nat] :
      ( ( K4 != bot_bot_set_nat )
     => pp(aa_set_nat_bool(member_nat(aa_set_nat_nat(complete_Inf_Inf_nat,K4)),K4)) ) ).

% Inf_nat_def1
tff(fact_9532_nat__not__finite,axiom,
    ~ pp(aa_set_nat_bool(finite_finite_nat,top_top_set_nat)) ).

% nat_not_finite
tff(fact_9533_infinite__UNIV__nat,axiom,
    ~ pp(aa_set_nat_bool(finite_finite_nat,top_top_set_nat)) ).

% infinite_UNIV_nat
tff(fact_9534_UN__lessThan__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_lessThan_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_lessThan_UNIV
tff(fact_9535_UN__atMost__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atMost_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atMost_UNIV
tff(fact_9536_range__enumerate,axiom,
    ! [S: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( image_nat_nat(infini8530281810654367211te_nat(S),top_top_set_nat) = S ) ) ).

% range_enumerate
tff(fact_9537_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_9538_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uo,N),top_top_set_nat) = set_or4665077453230672383an_nat(zero_zero_nat,N) ) ) ).

% range_mod
tff(fact_9539_suminf__eq__SUP__real,axiom,
    ! [X6: fun_nat_real] :
      ( summable_real(X6)
     => ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(X6,I2)))
       => ( suminf_real(X6) = comple1385675409528146559p_real(image_nat_real(aTP_Lamm_up(X6),top_top_set_nat)) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_9540_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_9541_range__mult,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero_real )
       => ( image_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),top_top_set_real) = insert_real(zero_zero_real,bot_bot_set_real) ) )
      & ( ( A2 != zero_zero_real )
       => ( image_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),top_top_set_real) = top_top_set_real ) ) ) ).

% range_mult
tff(fact_9542_root__def,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = zero_zero_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_uq,N)),X) ) ) ) ).

% root_def
tff(fact_9543_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9544_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D4: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( X != zero_zero_real )
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
             => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
               => ( D4 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
               => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),D4,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9545_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => has_fi5821293074295781190e_real(aTP_Lamm_ur,suminf_real(aa_real_fun_nat_real(aTP_Lamm_us,X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9546_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,X: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)),S))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D6))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_9547_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,X: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)),S))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D6))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_9548_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)),S))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D6))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_9549_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)),S))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D6))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_9550_deriv__nonneg__imp__mono,axiom,
    ! [A2: real,B3: real,G3: fun_real_real,G4: fun_real_real] :
      ( ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),set_or1222579329274155063t_real(A2,B3)))
         => has_fi5821293074295781190e_real(G3,aa_real_real(G4,X5),topolo2177554685111907308n_real(X5,top_top_set_real)) )
     => ( ! [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),set_or1222579329274155063t_real(A2,B3)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(G4,X5))) )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(G3,A2)),aa_real_real(G3,B3))) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_9551_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A2: real,B3: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X5))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),B3))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X5,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y4)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,A2)),aa_real_real(F,B3))) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_9552_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A2: real,B3: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X5))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),B3))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X5,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),zero_zero_real)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,B3)),aa_real_real(F,A2))) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_9553_DERIV__neg__imp__decreasing,axiom,
    ! [A2: real,B3: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X5))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),B3))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X5,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),zero_zero_real)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B3)),aa_real_real(F,A2))) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_9554_DERIV__pos__imp__increasing,axiom,
    ! [A2: real,B3: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X5))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),B3))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X5,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A2)),aa_real_real(F,B3))) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_9555_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D6))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_9556_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D6))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_9557_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D6))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_9558_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D6))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_9559_MVT2,axiom,
    ! [A2: real,B3: real,F: fun_real_real,F5: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X5))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),B3))
             => has_fi5821293074295781190e_real(F,aa_real_real(F5,X5),topolo2177554685111907308n_real(X5,top_top_set_real)) ) )
       => ? [Z3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),Z3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B3))
            & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B3)),aa_real_real(F,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),A2)),aa_real_real(F5,Z3)) ) ) ) ) ).

% MVT2
tff(fact_9560_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D))
             => ( aa_real_real(F,X) = aa_real_real(F,Y3) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_9561_DERIV__ln,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,X),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_9562_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X)),aa_real_real(F,Y3))) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_9563_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,Y3)),aa_real_real(F,X))) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_9564_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_9565_DERIV__pow,axiom,
    ! [N: nat,X: real,S2: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_ut,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(X,S2)) ).

% DERIV_pow
tff(fact_9566_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_uu,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_9567_DERIV__log,axiom,
    ! [X: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(log,B3),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,B3)),X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_log
tff(fact_9568_DERIV__fun__powr,axiom,
    ! [G3: fun_real_real,M: real,X: real,R2: real] :
      ( has_fi5821293074295781190e_real(G3,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G3,X)))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_uv(G3),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(G3,X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9569_DERIV__powr,axiom,
    ! [G3: fun_real_real,M: real,X: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G3,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G3,X)))
       => ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(X,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_uw(G3),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G3,X)),aa_real_real(F,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(ln_ln_real,aa_real_real(G3,X)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),aa_real_real(F,X))),aa_real_real(G3,X)))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_9570_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9571_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D4: real] :
      ( ( X != zero_zero_real )
     => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( D4 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
           => ( D4 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D4,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9572_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A3: set_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A3)) ) ).

% arcosh_real_has_field_derivative
tff(fact_9573_artanh__real__has__field__derivative,axiom,
    ! [X: real,A3: set_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X,A3)) ) ).

% artanh_real_has_field_derivative
tff(fact_9574_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9575_DERIV__arccos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => has_fi5821293074295781190e_real(arccos,aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_9576_DERIV__arcsin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_9577_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,N: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M4: nat,X5: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),X5),topolo2177554685111907308n_real(X5,top_top_set_real))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ux(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9578_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,N: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M4: nat,X5: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),X5),topolo2177554685111907308n_real(X5,top_top_set_real)) )
     => ? [T6: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
          & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ux(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9579_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( ( X != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9580_Maclaurin,axiom,
    ! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M4: nat,T6: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),H)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
           => ? [T6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),H))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_uy(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9581_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M4: nat,T6: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),H)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ? [T6: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),H))
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_uy(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9582_Maclaurin__minus,axiom,
    ! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),zero_zero_real))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M4: nat,T6: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),T6))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),zero_zero_real)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
           => ? [T6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),zero_zero_real))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_uy(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9583_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( X != zero_zero_real )
         => ( ! [M4: nat,X5: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),X5),topolo2177554685111907308n_real(X5,top_top_set_real))
           => ? [T6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T6)))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
                & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ux(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9584_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M4: nat,T6: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X))) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ux(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9585_Taylor,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A2: real,B3: real,C: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M4: nat,T6: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),B3)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),B3))
                 => ( ( X != C )
                   => ? [T6: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T6))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),C)) ) )
                        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T6))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),X)) ) )
                        & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_uz(Diff),C),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),C)),N))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_9586_Taylor__up,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A2: real,B3: real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M4: nat,T6: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),B3)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B3))
             => ? [T6: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T6))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),B3))
                  & ( aa_real_real(F,B3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_va(Diff),B3),C)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),C)),N))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_9587_Taylor__down,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A2: real,B3: real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M4: nat,T6: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),B3)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B3))
             => ? [T6: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),T6))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),C))
                  & ( aa_real_real(F,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_va(Diff),A2),C)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_9588_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B2: real] :
      ( ! [M4: nat,T6: real] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),H)) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
     => ( ( N = aa_nat_nat(suc,K) )
       => ! [M2: nat,T7: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T7))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T7),H)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_vc(N),Diff),B2),M2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T7)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_vd(Diff),M2),T7)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,T7),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T7,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9589_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R)))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_ve(F),X5)) )
     => ( pp(aa_set_real_bool(member_real(X0),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R))
         => has_fi5821293074295781190e_real(aTP_Lamm_vg(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_ve(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9590_DERIV__isconst3,axiom,
    ! [A2: real,B3: real,X: real,Y: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( pp(aa_set_real_bool(member_real(X),set_or1633881224788618240n_real(A2,B3)))
       => ( pp(aa_set_real_bool(member_real(Y),set_or1633881224788618240n_real(A2,B3)))
         => ( ! [X5: real] :
                ( pp(aa_set_real_bool(member_real(X5),set_or1633881224788618240n_real(A2,B3)))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X5,top_top_set_real)) )
           => ( aa_real_real(F,X) = aa_real_real(F,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_9591_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F5: fun_re4238438383274954274t_real,X0: real,A2: real,B3: real,L5: fun_nat_real] :
      ( ! [N2: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_vh(F),N2),aa_nat_real(aa_real_fun_nat_real(F5,X0),N2),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),set_or1633881224788618240n_real(A2,B3)))
           => summable_real(aa_real_fun_nat_real(F,X5)) )
       => ( pp(aa_set_real_bool(member_real(X0),set_or1633881224788618240n_real(A2,B3)))
         => ( summable_real(aa_real_fun_nat_real(F5,X0))
           => ( summable_real(L5)
             => ( ! [N2: nat,X5: real,Y3: real] :
                    ( pp(aa_set_real_bool(member_real(X5),set_or1633881224788618240n_real(A2,B3)))
                   => ( pp(aa_set_real_bool(member_real(Y3),set_or1633881224788618240n_real(A2,B3)))
                     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(F,X5),N2)),aa_nat_real(aa_real_fun_nat_real(F,Y3),N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(L5,N2)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X5),Y3))))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_vi(F),suminf_real(aa_real_fun_nat_real(F5,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_9592_Gcd__eq__Max,axiom,
    ! [M5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M5))
     => ( ( M5 != bot_bot_set_nat )
       => ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M5))
         => ( aa_set_nat_nat(gcd_Gcd_nat,M5) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_vj,M5))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_9593_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,set_or5834768355832116004an_nat(L,U))) ).

% finite_greaterThanLessThan
tff(fact_9594_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,set_or5832277885323065728an_int(L,U))) ).

% finite_greaterThanLessThan_int
tff(fact_9595_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_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_9596_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_tn,N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_9597_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or5832277885323065728an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int))) ).

% card_greaterThanLessThan_int
tff(fact_9598_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,L),U) = set_or5834768355832116004an_nat(L,U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_9599_isCont__Lb__Ub,axiom,
    ! [A2: real,B3: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( ! [X5: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X5))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),B3)) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X5,top_top_set_real),F) )
       => ? [L6: real,M9: real] :
            ( ! [X4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B3)) )
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L6),aa_real_real(F,X4)))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),M9)) ) )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L6),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),M9)) )
               => ? [X5: real] :
                    ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X5))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),B3))
                    & ( aa_real_real(F,X5) = Y4 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_9600_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X4))),R3)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,X4))) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_9601_LIM__fun__not__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( ( L != zero_zero_real )
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X4))),R3)) )
               => ( aa_real_real(F,X4) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_9602_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X4))),R3)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),zero_zero_real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_9603_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or5832277885323065728an_int(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_9604_isCont__inverse__function2,axiom,
    ! [A2: real,X: real,B3: real,G3: fun_real_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B3))
       => ( ! [Z3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),Z3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z3),B3))
               => ( aa_real_real(G3,aa_real_real(F,Z3)) = Z3 ) ) )
         => ( ! [Z3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),Z3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z3),B3))
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G3) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_9605_Sup__nat__def,axiom,
    ! [X6: set_nat] :
      ( ( ( X6 = bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X6) = zero_zero_nat ) )
      & ( ( X6 != bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X6) = aa_set_nat_nat(lattic8265883725875713057ax_nat,X6) ) ) ) ).

% Sup_nat_def
tff(fact_9606_card__le__Suc__Max,axiom,
    ! [S: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S)))) ) ).

% card_le_Suc_Max
tff(fact_9607_divide__nat__def,axiom,
    ! [N: nat,M: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_vk(N),M))) ) ) ) ).

% divide_nat_def
tff(fact_9608_isCont__arcosh,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_9609_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D4: real,G3: fun_real_real,X: real,A2: real,B3: real] :
      ( has_fi5821293074295781190e_real(F,D4,topolo2177554685111907308n_real(aa_real_real(G3,X),top_top_set_real))
     => ( ( D4 != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B3))
           => ( ! [Y3: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),Y3))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),B3))
                   => ( aa_real_real(F,aa_real_real(G3,Y3)) = Y3 ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),G3)
               => has_fi5821293074295781190e_real(G3,aa_real_real(inverse_inverse_real,D4),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_9610_isCont__arccos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_9611_isCont__arcsin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_9612_LIM__less__bound,axiom,
    ! [B3: real,X: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),X))
     => ( ! [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),set_or1633881224788618240n_real(B3,X)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X5))) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X))) ) ) ) ).

% LIM_less_bound
tff(fact_9613_isCont__artanh,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_9614_isCont__inverse__function,axiom,
    ! [D: real,X: real,G3: fun_real_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
     => ( ! [Z3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z3),X))),D))
           => ( aa_real_real(G3,aa_real_real(F,Z3)) = Z3 ) )
       => ( ! [Z3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z3),X))),D))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G3) ) ) ) ).

% isCont_inverse_function
tff(fact_9615_GMVT_H,axiom,
    ! [A2: real,B3: real,F: fun_real_real,G3: fun_real_real,G4: fun_real_real,F5: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( ! [Z3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),Z3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z3),B3))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) ) )
       => ( ! [Z3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),Z3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z3),B3))
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),G3) ) )
         => ( ! [Z3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),Z3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B3))
                 => has_fi5821293074295781190e_real(G3,aa_real_real(G4,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
           => ( ! [Z3: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),Z3))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B3))
                   => has_fi5821293074295781190e_real(F,aa_real_real(F5,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
             => ? [C3: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C3),B3))
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B3)),aa_real_real(F,A2))),aa_real_real(G4,C3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G3,B3)),aa_real_real(G3,A2))),aa_real_real(F5,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_9616_summable__Leibniz_I3_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A2)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(A2,zero_zero_nat)),zero_zero_real))
         => ! [N5: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_vl(A2))),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vl(A2)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)),one_one_nat))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vl(A2)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_9617_summable__Leibniz_I2_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A2)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(A2,zero_zero_nat)))
         => ! [N5: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_vl(A2))),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vl(A2)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vl(A2)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)),one_one_nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_9618_summable__Leibniz_H_I4_J,axiom,
    ! [A2: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N2))),aa_nat_real(A2,N2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(aTP_Lamm_vl(A2))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vl(A2)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_9619_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
tff(fact_9620_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vm,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_9621_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_9622_monoseq__convergent,axiom,
    ! [X6: fun_nat_real,B2: real] :
      ( topolo6980174941875973593q_real(X6)
     => ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(X6,I2))),B2))
       => ~ ! [L6: real] : ~ filterlim_nat_real(X6,topolo2815343760600316023s_real(L6),at_top_nat) ) ) ).

% monoseq_convergent
tff(fact_9623_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G3: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(G3,aa_nat_nat(suc,N2))),aa_nat_real(G3,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G3,N2)))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vn(F),G3),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L4: real] :
                ( ! [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N5)),L4))
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L4),at_top_nat)
                & ! [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L4),aa_nat_real(G3,N5)))
                & filterlim_nat_real(G3,topolo2815343760600316023s_real(L4),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_9624_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun_nat_real] :
      ( ! [R3: real] :
        ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R3),aa_nat_real(X6,N2))) )
     => filterlim_nat_real(aTP_Lamm_vo(X6),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9625_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_vp,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_9626_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),L))
       => ( ! [E: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
             => ? [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N5)),E))) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_9627_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_9628_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_vq(X),A2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_9629_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real))
     => filterlim_nat_real(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_9630_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real))
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_9631_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_vr,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_9632_summable,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N2))),aa_nat_real(A2,N2)))
         => summable_real(aTP_Lamm_vl(A2)) ) ) ) ).

% summable
tff(fact_9633_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_gt,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9634_summable__Leibniz_H_I3_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N2))),aa_nat_real(A2,N2)))
         => filterlim_nat_real(aTP_Lamm_vs(A2),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_vl(A2))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_9635_summable__Leibniz_H_I2_J,axiom,
    ! [A2: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N2))),aa_nat_real(A2,N2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vl(A2)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),suminf_real(aTP_Lamm_vl(A2)))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_9636_sums__alternating__upper__lower,axiom,
    ! [A2: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N2))),aa_nat_real(A2,N2)))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N2)))
       => ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L4: real] :
              ( ! [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vl(A2)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)))),L4))
              & filterlim_nat_real(aTP_Lamm_vs(A2),topolo2815343760600316023s_real(L4),at_top_nat)
              & ! [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L4),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vl(A2)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)),one_one_nat)))))
              & filterlim_nat_real(aTP_Lamm_vt(A2),topolo2815343760600316023s_real(L4),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9637_summable__Leibniz_H_I5_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N2))),aa_nat_real(A2,N2)))
         => filterlim_nat_real(aTP_Lamm_vt(A2),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_vl(A2))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9638_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B3: real,F: fun_real_real,Flim: real] :
      ( ! [X5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),X5))
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X5,top_top_set_real))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),zero_zero_real)) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Flim),aa_real_real(F,B3))) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_9639_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: fun_real_real,F2: filter_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( filterlim_real_real(F,at_bot_real,F2)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vu(N),F),at_top_real,F2) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_9640_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B3: real,F: fun_real_real,Flim: real] :
      ( ! [X5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),B3))
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X5,top_top_set_real))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4)) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Flim),aa_real_real(F,B3))) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_9641_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: fun_real_real,F2: filter_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( filterlim_real_real(F,at_bot_real,F2)
       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vu(N),F),at_bot_real,F2) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_9642_GMVT,axiom,
    ! [A2: real,B3: real,F: fun_real_real,G3: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( ! [X5: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X5))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),B3)) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X5,top_top_set_real),F) )
       => ( ! [X5: real] :
              ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X5))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),B3)) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X5,top_top_set_real)) )
         => ( ! [X5: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X5))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),B3)) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X5,top_top_set_real),G3) )
           => ( ! [X5: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X5))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),B3)) )
                 => differ6690327859849518006l_real(G3,topolo2177554685111907308n_real(X5,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C3: real] :
                  ( has_fi5821293074295781190e_real(G3,G_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C3),B3))
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B3)),aa_real_real(F,A2))),G_c) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G3,B3)),aa_real_real(G3,A2))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_9643_at__bot__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_bot_real,at_infinity_real) ).

% at_bot_le_at_infinity
tff(fact_9644_at__top__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_top_real,at_infinity_real) ).

% at_top_le_at_infinity
tff(fact_9645_MVT,axiom,
    ! [A2: real,B3: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B3),F)
       => ( ! [X5: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X5))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),B3))
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X5,top_top_set_real)) ) )
         => ? [L4: real,Z3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),Z3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B3))
              & has_fi5821293074295781190e_real(F,L4,topolo2177554685111907308n_real(Z3,top_top_set_real))
              & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B3)),aa_real_real(F,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),A2)),L4) ) ) ) ) ) ).

% MVT
tff(fact_9646_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_bool] :
      ( ! [X5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),X5))
         => pp(aa_nat_bool(P,X5)) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_9647_eventually__sequentially,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N9: nat] :
        ! [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N3))
         => pp(aa_nat_bool(P,N3)) ) ) ).

% eventually_sequentially
tff(fact_9648_le__sequentially,axiom,
    ! [F2: filter_nat] :
      ( ord_le2510731241096832064er_nat(F2,at_top_nat)
    <=> ! [N9: nat] : eventually_nat(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),F2) ) ).

% le_sequentially
tff(fact_9649_continuous__on__arcosh_H,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A3,F)
     => ( ! [X5: real] :
            ( pp(aa_set_real_bool(member_real(X5),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,X5))) )
       => topolo5044208981011980120l_real(A3,aTP_Lamm_vv(F)) ) ) ).

% continuous_on_arcosh'
tff(fact_9650_eventually__at__left__real,axiom,
    ! [B3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A2))
     => eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_vw(B3),A2),topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2))) ) ).

% eventually_at_left_real
tff(fact_9651_continuous__image__closed__interval,axiom,
    ! [A2: real,B3: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B3))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B3),F)
       => ? [C3: real,D6: real] :
            ( ( image_real_real(F,set_or1222579329274155063t_real(A2,B3)) = set_or1222579329274155063t_real(C3,D6) )
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C3),D6)) ) ) ) ).

% continuous_image_closed_interval
tff(fact_9652_continuous__on__artanh,axiom,
    ! [A3: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)))
     => topolo5044208981011980120l_real(A3,artanh_real) ) ).

% continuous_on_artanh
tff(fact_9653_Rolle__deriv,axiom,
    ! [A2: real,B3: real,F: fun_real_real,F5: fun_re7681423839302404990l_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( ( aa_real_real(F,A2) = aa_real_real(F,B3) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B3),F)
         => ( ! [X5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X5))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),B3))
                 => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F5,X5),topolo2177554685111907308n_real(X5,top_top_set_real)) ) )
           => ? [Z3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),Z3))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B3))
                & ! [X4: real] : aa_real_real(aa_rea7340605660300375871l_real(F5,Z3),X4) = zero_zero_real ) ) ) ) ) ).

% Rolle_deriv
tff(fact_9654_Bseq__eq__bounded,axiom,
    ! [F: fun_nat_real,A2: real,B3: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(image_nat_real(F,top_top_set_nat)),set_or1222579329274155063t_real(A2,B3)))
     => bfun_nat_real(F,at_top_nat) ) ).

% Bseq_eq_bounded
tff(fact_9655_mvt,axiom,
    ! [A2: real,B3: real,F: fun_real_real,F5: fun_re7681423839302404990l_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B3),F)
       => ( ! [X5: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X5))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),B3))
               => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F5,X5),topolo2177554685111907308n_real(X5,top_top_set_real)) ) )
         => ~ ! [Xi: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),Xi))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xi),B3))
                 => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B3)),aa_real_real(F,A2)) != aa_real_real(aa_rea7340605660300375871l_real(F5,Xi),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),A2)) ) ) ) ) ) ) ).

% mvt
tff(fact_9656_Bseq__realpow,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => bfun_nat_real(aa_real_fun_nat_real(power_power_real,X),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_9657_DERIV__pos__imp__increasing__open,axiom,
    ! [A2: real,B3: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X5))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),B3))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X5,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4)) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B3),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A2)),aa_real_real(F,B3))) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_9658_DERIV__neg__imp__decreasing__open,axiom,
    ! [A2: real,B3: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X5))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),B3))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X5,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),zero_zero_real)) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B3),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B3)),aa_real_real(F,A2))) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_9659_DERIV__isconst__end,axiom,
    ! [A2: real,B3: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B3),F)
       => ( ! [X5: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X5))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),B3))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X5,top_top_set_real)) ) )
         => ( aa_real_real(F,B3) = aa_real_real(F,A2) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_9660_DERIV__isconst2,axiom,
    ! [A2: real,B3: real,F: fun_real_real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B3),F)
       => ( ! [X5: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X5))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),B3))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X5,top_top_set_real)) ) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),B3))
             => ( aa_real_real(F,X) = aa_real_real(F,A2) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_9661_Rolle,axiom,
    ! [A2: real,B3: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => ( ( aa_real_real(F,A2) = aa_real_real(F,B3) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B3),F)
         => ( ! [X5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X5))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),B3))
                 => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X5,top_top_set_real)) ) )
           => ? [Z3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),Z3))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B3))
                & has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(Z3,top_top_set_real)) ) ) ) ) ) ).

% Rolle
tff(fact_9662_eventually__at__right__real,axiom,
    ! [A2: real,B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B3))
     => eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_vw(A2),B3),topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2))) ) ).

% eventually_at_right_real
tff(fact_9663_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_9664_greaterThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(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_9665_INT__greaterThan__UNIV,axiom,
    comple7806235888213564991et_nat(image_nat_set_nat(set_or1210151606488870762an_nat,top_top_set_nat)) = bot_bot_set_nat ).

% INT_greaterThan_UNIV
tff(fact_9666_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,set_or6659071591806873216st_nat(L,U))) ).

% finite_greaterThanAtMost
tff(fact_9667_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,set_or6659071591806873216st_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_greaterThanAtMost
tff(fact_9668_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,L),U) = set_or6659071591806873216st_nat(L,U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_9669_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_bool,B3: nat] :
      ( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B3)) )
       => pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).

% GreatestI_ex_nat
tff(fact_9670_Greatest__le__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B3: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B3)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),order_Greatest_nat(P))) ) ) ).

% Greatest_le_nat
tff(fact_9671_GreatestI__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B3: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B3)) )
       => pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).

% GreatestI_nat
tff(fact_9672_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,set_or6656581121297822940st_int(L,U))) ).

% finite_greaterThanAtMost_int
tff(fact_9673_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_9674_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or6656581121297822940st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).

% card_greaterThanAtMost_int
tff(fact_9675_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_or1210151606488870762an_nat,K) ).

% atLeast_Suc_greaterThan
tff(fact_9676_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or6656581121297822940st_int(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_9677_decseq__bounded,axiom,
    ! [X6: fun_nat_real,B2: real] :
      ( order_9091379641038594480t_real(X6)
     => ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_nat_real(X6,I2)))
       => bfun_nat_real(X6,at_top_nat) ) ) ).

% decseq_bounded
tff(fact_9678_decseq__convergent,axiom,
    ! [X6: fun_nat_real,B2: real] :
      ( order_9091379641038594480t_real(X6)
     => ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_nat_real(X6,I2)))
       => ~ ! [L6: real] :
              ( filterlim_nat_real(X6,topolo2815343760600316023s_real(L6),at_top_nat)
             => ~ ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L6),aa_nat_real(X6,I3))) ) ) ) ).

% decseq_convergent
tff(fact_9679_UN__atLeast__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atLeast_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atLeast_UNIV
tff(fact_9680_atLeast__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_atLeast_nat,K)),insert_nat(K,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_9681_continuous__on__arcosh,axiom,
    ! [A3: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),set_ord_atLeast_real(one_one_real)))
     => topolo5044208981011980120l_real(A3,arcosh_real) ) ).

% continuous_on_arcosh
tff(fact_9682_bdd__above__nat,axiom,
    ! [X6: set_nat] :
      ( pp(aa_set_nat_bool(condit2214826472909112428ve_nat,X6))
    <=> pp(aa_set_nat_bool(finite_finite_nat,X6)) ) ).

% bdd_above_nat
tff(fact_9683_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( pp(Y)
          <=> ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
             => ( pp(Y)
              <=> ~ ( ( Deg2 = Xa2 )
                    & ! [X3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
                       => pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                    & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_vx(Deg2),TreeList),Summary2))),Mima)) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9684_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                     => pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                  & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                  & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                  & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_vx(Deg2),TreeList),Summary2))),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9685_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
             => ( ( Deg2 = Xa2 )
                & ! [X5: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList)))
                   => pp(aa_nat_bool(vEBT_VEBT_valid(X5),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_vx(Deg2),TreeList),Summary2))),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9686_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList2,Summary)),Deg3))
    <=> ( ( Deg = Deg3 )
        & ! [X3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
           => pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
        & pp(aa_nat_bool(vEBT_VEBT_valid(Summary),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
        & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
        & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_vx(Deg),TreeList2),Summary))),Mima2)) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9687_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2))
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList,Summary2),Xa2))
                 => ( ( Deg2 = Xa2 )
                    & ! [X5: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList)))
                       => pp(aa_nat_bool(vEBT_VEBT_valid(X5),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                    & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_vx(Deg2),TreeList),Summary2))),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9688_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2))
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList,Summary2),Xa2))
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X4: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                         => pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                      & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_vx(Deg2),TreeList),Summary2))),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9689_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( pp(Y)
                <=> ( Xa2 = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
               => ( ( pp(Y)
                  <=> ( ( Deg2 = Xa2 )
                      & ! [X3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
                         => pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                      & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_vx(Deg2),TreeList),Summary2))),Mima)) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList,Summary2),Xa2)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_9690_Sup__real__def,axiom,
    ! [X6: set_real] : comple1385675409528146559p_real(X6) = ord_Least_real(aTP_Lamm_vy(X6)) ).

% Sup_real_def
tff(fact_9691_Sup__int__def,axiom,
    ! [X6: set_int] : complete_Sup_Sup_int(X6) = the_int(aTP_Lamm_vz(X6)) ).

% Sup_int_def
tff(fact_9692_uniformity__real__def,axiom,
    topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_wb,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_real_def
tff(fact_9693_uniformity__complex__def,axiom,
    topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_wd,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_complex_def
tff(fact_9694_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( code_bit_cut_integer(K) = produc5135875973608410785r_bool(zero_z3403309356797280102nteger,fFalse) ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( code_bit_cut_integer(K) = produc6117575242473779287r_bool(aTP_Lamm_we(K),code_divmod_abs(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).

% bit_cut_integer_code
tff(fact_9695_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
           => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K))
               => ( code_divmod_integer(K,L) = code_divmod_abs(K,L) ) )
              & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K))
               => ( code_divmod_integer(K,L) = produc6916734918728496179nteger(aTP_Lamm_wf(L),code_divmod_abs(K,L)) ) ) ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
           => ( ( ( L = zero_z3403309356797280102nteger )
               => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,K) ) )
              & ( ( L != zero_z3403309356797280102nteger )
               => ( code_divmod_integer(K,L) = produc6499014454317279255nteger(uminus1351360451143612070nteger,if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_wg(L),code_divmod_abs(K,L)))) ) ) ) ) ) ) ) ).

% divmod_integer_code
tff(fact_9696_eventually__prod__sequentially,axiom,
    ! [P: fun_Pr7757163693017731481t_bool] :
      ( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N9: nat] :
        ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),M3))
         => ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N3))
             => pp(aa_Pro2232559814647311250t_bool(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N3),M3))) ) ) ) ).

% eventually_prod_sequentially
tff(fact_9697_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_9698_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se8568078237143864401it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9699_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se8568078237143864401it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% drop_bit_negative_int_iff
tff(fact_9700_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% drop_bit_of_Suc_0
tff(fact_9701_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_9702_mono__times__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N)) ) ).

% mono_times_nat
tff(fact_9703_incseq__bounded,axiom,
    ! [X6: fun_nat_real,B2: real] :
      ( order_mono_nat_real(X6)
     => ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,I2)),B2))
       => bfun_nat_real(X6,at_top_nat) ) ) ).

% incseq_bounded
tff(fact_9704_incseq__convergent,axiom,
    ! [X6: fun_nat_real,B2: real] :
      ( order_mono_nat_real(X6)
     => ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,I2)),B2))
       => ~ ! [L6: real] :
              ( filterlim_nat_real(X6,topolo2815343760600316023s_real(L6),at_top_nat)
             => ~ ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,I3)),L6)) ) ) ) ).

% incseq_convergent
tff(fact_9705_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S))
    <=> ~ pp(aa_set_nat_bool(finite_finite_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),S))) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_9706_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wh,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_9707_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_9708_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F: fun_nat_real,G3: fun_nat_nat] :
      ( ! [X5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X5)))
     => ( order_mono_nat_real(F)
       => ( order_5726023648592871131at_nat(G3)
         => ( bfun_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_wi(F),G3),at_top_nat)
          <=> bfun_nat_real(F,at_top_nat) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
tff(fact_9709_strict__mono__imp__increasing,axiom,
    ! [F: fun_nat_nat,N: nat] :
      ( order_5726023648592871131at_nat(F)
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(F,N))) ) ).

% strict_mono_imp_increasing
tff(fact_9710_infinite__enumerate,axiom,
    ! [S: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
     => ? [R3: fun_nat_nat] :
          ( order_5726023648592871131at_nat(R3)
          & ! [N5: nat] : pp(aa_set_nat_bool(member_nat(aa_nat_nat(R3,N5)),S)) ) ) ).

% infinite_enumerate
tff(fact_9711_strict__mono__enumerate,axiom,
    ! [S: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
     => order_5726023648592871131at_nat(infini8530281810654367211te_nat(S)) ) ).

% strict_mono_enumerate
tff(fact_9712_take__bit__num__def,axiom,
    ! [N: nat,M: num] :
      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) = zero_zero_nat )
       => ( bit_take_bit_num(N,M) = none_num ) )
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) != zero_zero_nat )
       => ( bit_take_bit_num(N,M) = some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)))) ) ) ) ).

% take_bit_num_def
tff(fact_9713_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9714_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_9715_num__of__nat__One,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),one_one_nat))
     => ( aa_nat_num(num_of_nat,N) = one ) ) ).

% num_of_nat_One
tff(fact_9716_num__of__nat__double,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)) = aa_num_num(bit0,aa_nat_num(num_of_nat,N)) ) ) ).

% num_of_nat_double
tff(fact_9717_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,M)),aa_nat_num(num_of_nat,N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9718_pos__deriv__imp__strict__mono,axiom,
    ! [F: fun_real_real,F5: fun_real_real] :
      ( ! [X5: real] : has_fi5821293074295781190e_real(F,aa_real_real(F5,X5),topolo2177554685111907308n_real(X5,top_top_set_real))
     => ( ! [X5: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F5,X5)))
       => order_7092887310737990675l_real(F) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_9719_Suc__funpow,axiom,
    ! [N: nat] : compow_fun_nat_nat(N,suc) = aa_nat_fun_nat_nat(plus_plus_nat,N) ).

% Suc_funpow
tff(fact_9720_inj__sgn__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_real_bool(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_uq,N)),top_top_set_real)) ) ).

% inj_sgn_power
tff(fact_9721_log__inj,axiom,
    ! [B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B3))
     => pp(aa_set_real_bool(inj_on_real_real(aa_rea7340605660300375871l_real(log,B3)),set_or5849166863359141190n_real(zero_zero_real))) ) ).

% log_inj
tff(fact_9722_inj__Suc,axiom,
    ! [N4: set_nat] : pp(aa_set_nat_bool(inj_on_nat_nat(suc),N4)) ).

% inj_Suc
tff(fact_9723_inj__on__diff__nat,axiom,
    ! [N4: set_nat,K: nat] :
      ( ! [N2: nat] :
          ( pp(aa_set_nat_bool(member_nat(N2),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2)) )
     => pp(aa_set_nat_bool(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ul,K)),N4)) ) ).

% inj_on_diff_nat
tff(fact_9724_inj__on__set__encode,axiom,
    pp(aa_set_set_nat_bool(inj_on_set_nat_nat(nat_set_encode),collect_set_nat(finite_finite_nat))) ).

% inj_on_set_encode
tff(fact_9725_summable__reindex,axiom,
    ! [F: fun_nat_real,G3: fun_nat_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(inj_on_nat_nat(G3),top_top_set_nat))
       => ( ! [X5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X5)))
         => summable_real(comp_nat_real_nat(F,G3)) ) ) ) ).

% summable_reindex
tff(fact_9726_suminf__reindex__mono,axiom,
    ! [F: fun_nat_real,G3: fun_nat_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(inj_on_nat_nat(G3),top_top_set_nat))
       => ( ! [X5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X5)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(comp_nat_real_nat(F,G3))),suminf_real(F))) ) ) ) ).

% suminf_reindex_mono
tff(fact_9727_suminf__reindex,axiom,
    ! [F: fun_nat_real,G3: fun_nat_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(inj_on_nat_nat(G3),top_top_set_nat))
       => ( ! [X5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X5)))
         => ( ! [X5: nat] :
                ( ~ pp(aa_set_nat_bool(member_nat(X5),image_nat_nat(G3,top_top_set_nat)))
               => ( aa_nat_real(F,X5) = zero_zero_real ) )
           => ( suminf_real(comp_nat_real_nat(F,G3)) = suminf_real(F) ) ) ) ) ) ).

% suminf_reindex
tff(fact_9728_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_bl,aTP_Lamm_bk) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9729_push__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% push_bit_of_Suc_0
tff(fact_9730_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se545348938243370406it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% push_bit_nonnegative_int_iff
tff(fact_9731_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se545348938243370406it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% push_bit_negative_int_iff
tff(fact_9732_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_se545348938243370406it_int(M),K)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_int
tff(fact_9733_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q3)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9734_powr__real__of__int_H,axiom,
    ! [X: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( ( X != zero_zero_real )
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N)) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,N)) = aa_int_real(power_int_real(X),N) ) ) ) ).

% powr_real_of_int'
tff(fact_9735_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,I)),J))
     => ( linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9736_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),J))
     => ( linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9737_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9738_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I))))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9739_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = aa_num_num(inc,aa_nat_num(num_of_nat,N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = one ) ) ) ).

% num_of_nat.simps(2)
tff(fact_9740_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_wj) ).

% Rats_eq_int_div_nat
tff(fact_9741_rat__floor__lemma,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3))),aa_int_rat(fract(A2),B3)))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(fract(A2),B3)),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)),one_one_int)))) ) ).

% rat_floor_lemma
tff(fact_9742_less__rat,axiom,
    ! [B3: int,D: int,A2: int,C: int] :
      ( ( B3 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(fract(A2),B3)),aa_int_rat(fract(C),D)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B3)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),D)))) ) ) ) ).

% less_rat
tff(fact_9743_le__rat,axiom,
    ! [B3: int,D: int,A2: int,C: int] :
      ( ( B3 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(fract(A2),B3)),aa_int_rat(fract(C),D)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B3)),aa_int_int(aa_int_fun_int_int(times_times_int,B3),D)))) ) ) ) ).

% le_rat
tff(fact_9744_Rats__dense__in__real,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),field_5140801741446780682s_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X5))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Y)) ) ) ).

% Rats_dense_in_real
tff(fact_9745_Rats__no__bot__less,axiom,
    ! [X: real] :
    ? [X5: real] :
      ( pp(aa_set_real_bool(member_real(X5),field_5140801741446780682s_real))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),X)) ) ).

% Rats_no_bot_less
tff(fact_9746_Rat__induct__pos,axiom,
    ! [P: fun_rat_bool,Q3: rat] :
      ( ! [A: int,B: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
         => pp(aa_rat_bool(P,aa_int_rat(fract(A),B))) )
     => pp(aa_rat_bool(P,Q3)) ) ).

% Rat_induct_pos
tff(fact_9747_Rats__no__top__le,axiom,
    ! [X: real] :
    ? [X5: real] :
      ( pp(aa_set_real_bool(member_real(X5),field_5140801741446780682s_real))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),X5)) ) ).

% Rats_no_top_le
tff(fact_9748_zero__less__Fract__iff,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(fract(A2),B3)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2)) ) ) ).

% zero_less_Fract_iff
tff(fact_9749_Fract__less__zero__iff,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(fract(A2),B3)),zero_zero_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ) ).

% Fract_less_zero_iff
tff(fact_9750_one__less__Fract__iff,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(fract(A2),B3)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),A2)) ) ) ).

% one_less_Fract_iff
tff(fact_9751_Fract__less__one__iff,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(fract(A2),B3)),one_one_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3)) ) ) ).

% Fract_less_one_iff
tff(fact_9752_zero__le__Fract__iff,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(fract(A2),B3)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2)) ) ) ).

% zero_le_Fract_iff
tff(fact_9753_Fract__le__zero__iff,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(fract(A2),B3)),zero_zero_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) ) ) ).

% Fract_le_zero_iff
tff(fact_9754_one__le__Fract__iff,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(fract(A2),B3)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A2)) ) ) ).

% one_le_Fract_iff
tff(fact_9755_Fract__le__one__iff,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(fract(A2),B3)),one_one_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3)) ) ) ).

% Fract_le_one_iff
tff(fact_9756_positive__rat,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_rat_bool(positive,aa_int_rat(fract(A2),B3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3))) ) ).

% positive_rat
tff(fact_9757_less__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> pp(aa_rat_bool(positive,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Y),X))) ) ).

% less_rat_def
tff(fact_9758_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)) ).

% min_Suc_Suc
tff(fact_9759_min__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N) = zero_zero_nat ).

% min_0L
tff(fact_9760_min__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9761_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)) ).

% nat_mult_min_right
tff(fact_9762_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) ).

% nat_mult_min_left
tff(fact_9763_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),I) ).

% min_diff
tff(fact_9764_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9765_Arg__bounded,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(arg,Z)),pi)) ) ).

% Arg_bounded
tff(fact_9766_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)) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z)))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(arg,Z)),pi)) ) ) ).

% Arg_correct
tff(fact_9767_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_complex_bool(bij_betw_nat_complex(aTP_Lamm_wk(N),aa_nat_set_nat(set_ord_lessThan_nat,N)),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_bs,N)))) ) ).

% bij_betw_roots_unity
tff(fact_9768_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,X) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
         => ( aa_complex_real(arg,Z) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_9769_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_set_complex_bool(bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,N),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,N))))),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_bs,N))),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_br(C),N)))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9770_bij__betw__Suc,axiom,
    ! [M5: set_nat,N4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(bij_betw_nat_nat(suc),M5),N4))
    <=> ( image_nat_nat(suc,M5) = N4 ) ) ).

% bij_betw_Suc
tff(fact_9771_bij__enumerate,axiom,
    ! [S: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(bij_betw_nat_nat(infini8530281810654367211te_nat(S)),top_top_set_nat),S)) ) ).

% bij_enumerate
tff(fact_9772_Arg__def,axiom,
    ! [Z: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_real(arg,Z) = zero_zero_real ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_real(arg,Z) = fChoice_real(aa_com8943241093633526360l_bool(aTP_Lamm_wl,Z)) ) ) ) ).

% Arg_def
tff(fact_9773_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_wn),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2))) ) ).

% less_eq_int.rep_eq
tff(fact_9774_less__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Xa2))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_wp),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2))) ) ).

% less_int.rep_eq
tff(fact_9775_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,abs_Integ(Xa2)),abs_Integ(X)))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_wn),Xa2),X)) ) ).

% less_eq_int.abs_eq
tff(fact_9776_zero__int__def,axiom,
    zero_zero_int = abs_Integ(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).

% zero_int_def
tff(fact_9777_int__def,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = abs_Integ(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N),zero_zero_nat)) ).

% int_def
tff(fact_9778_one__int__def,axiom,
    one_one_int = abs_Integ(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int_def
tff(fact_9779_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,abs_Integ(Xa2)),abs_Integ(X)))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_wp),Xa2),X)) ) ).

% less_int.abs_eq
tff(fact_9780_card__length__sum__list__rec,axiom,
    ! [M: nat,N4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
     => ( finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_wq(M),N4))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_wr(M),N4)))),finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_ws(M),N4)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9781_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( hd_nat(upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_9782_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( upt(I,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_9783_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I)) ) ) ).

% upt_eq_Nil_conv
tff(fact_9784_take__upt,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)),N))
     => ( take_nat(M,upt(I,N)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)) ) ) ).

% take_upt
tff(fact_9785_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J))
     => ( aa_nat_nat(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_9786_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N)) = cons_nat(aa_num_nat(numeral_numeral_nat,M),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N)) = nil_nat ) ) ) ).

% upt_rec_numeral
tff(fact_9787_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups4561878855575611511st_nat(upt(M,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_qb),set_or4665077453230672383an_nat(M,N)) ) ) ).

% sum_list_upt
tff(fact_9788_map__add__upt,axiom,
    ! [N: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wt,N),upt(zero_zero_nat,M)) = upt(N,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% map_add_upt
tff(fact_9789_atLeast__upt,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = set_nat2(upt(zero_zero_nat,N)) ).

% atLeast_upt
tff(fact_9790_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_9791_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),upt(M,N))) ).

% sorted_wrt_upt
tff(fact_9792_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_9793_sorted__upt,axiom,
    ! [M: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),upt(M,N))) ).

% sorted_upt
tff(fact_9794_atMost__upto,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,N))) ).

% atMost_upto
tff(fact_9795_upt__rec,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = nil_nat ) ) ) ).

% upt_rec
tff(fact_9796_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_9797_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_9798_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = nil_nat ) ) ) ).

% upt_Suc
tff(fact_9799_map__decr__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(aTP_Lamm_wu,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_9800_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(X,Xs) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( I = X )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9801_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Ns))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Ns)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I))) ) ) ).

% sorted_wrt_less_idx
tff(fact_9802_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : sorted_wrt_int(ord_less_int,upto(I,J)) ).

% sorted_wrt_upto
tff(fact_9803_sorted__upto,axiom,
    ! [M: int,N: int] : sorted_wrt_int(ord_less_eq_int,upto(M,N)) ).

% sorted_upto
tff(fact_9804_Suc__0__div__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_div_numeral
tff(fact_9805_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y = product_Pair_int_int(one_one_int,zero_zero_int) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y = product_Pair_int_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2))))) ) ) ) ) ).

% bezw.elims
tff(fact_9806_bezw_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( bezw(X,Y) = product_Pair_int_int(one_one_int,zero_zero_int) ) )
      & ( ( Y != zero_zero_nat )
       => ( bezw(X,Y) = product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ) ).

% bezw.simps
tff(fact_9807_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Y))
     => ( bezw(X,Y) = product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ).

% bezw_non_0
tff(fact_9808_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = product_Pair_int_int(one_one_int,zero_zero_int) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y = product_Pair_int_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2))))) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).

% bezw.pelims
tff(fact_9809_Suc__0__mod__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_mod_numeral
tff(fact_9810_Rat_Opositive_Orep__eq,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(positive,X))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(aa_rat7754553326175492643nt_int(rep_Rat,X))),product_snd_int_int(aa_rat7754553326175492643nt_int(rep_Rat,X))))) ) ).

% Rat.positive.rep_eq
tff(fact_9811_normalize__def,axiom,
    ! [P3: product_prod_int_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P3)))
       => ( normalize(P3) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P3)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3))),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P3)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P3)))
       => ( ( ( product_snd_int_int(P3) = zero_zero_int )
           => ( normalize(P3) = product_Pair_int_int(zero_zero_int,one_one_int) ) )
          & ( ( product_snd_int_int(P3) != zero_zero_int )
           => ( normalize(P3) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P3)),aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P3)),aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3))))) ) ) ) ) ) ).

% normalize_def
tff(fact_9812_gcd__pos__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N)))
    <=> ( ( M != zero_zero_int )
        | ( N != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_9813_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y))) ).

% gcd_ge_0_int
tff(fact_9814_gcd__le2__int,axiom,
    ! [B3: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A2),B3)),B3)) ) ).

% gcd_le2_int
tff(fact_9815_gcd__le1__int,axiom,
    ! [A2: int,B3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A2),B3)),A2)) ) ).

% gcd_le1_int
tff(fact_9816_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: fun_int_bool] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
         => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y))) ) )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,Y)))) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),Y))) ) )
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),aa_int_int(uminus_uminus_int,Y)))) ) )
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y))) ) ) ) ) ).

% gcd_cases_int
tff(fact_9817_gcd__unique__int,axiom,
    ! [D: int,A2: int,B3: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),D))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),A2))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),B3))
        & ! [E3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,E3),A2))
              & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,E3),B3)) )
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,E3),D)) ) )
    <=> ( D = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A2),B3) ) ) ).

% gcd_unique_int
tff(fact_9818_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Y))
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),Y)) ) ) ).

% gcd_non_0_int
tff(fact_9819_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),B3) = zero_zero_nat )
    <=> ( ( A2 = zero_zero_nat )
        & ( B3 = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_9820_gcd__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),A2) = A2 ).

% gcd_nat.left_neutral
tff(fact_9821_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B3: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),B3) )
    <=> ( ( A2 = zero_zero_nat )
        & ( B3 = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_9822_gcd__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),zero_zero_nat) = A2 ).

% gcd_nat.right_neutral
tff(fact_9823_gcd__0__nat,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),zero_zero_nat) = X ).

% gcd_0_nat
tff(fact_9824_gcd__0__left__nat,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),X) = X ).

% gcd_0_left_nat
tff(fact_9825_gcd__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_9826_gcd__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N)))
    <=> ( ( M != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9827_Gcd__in,axiom,
    ! [A3: set_nat] :
      ( ! [A: nat,B: nat] :
          ( pp(aa_set_nat_bool(member_nat(A),A3))
         => ( pp(aa_set_nat_bool(member_nat(B),A3))
           => pp(aa_set_nat_bool(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A3)) ) )
     => ( ( A3 != bot_bot_set_nat )
       => pp(aa_set_nat_bool(member_nat(aa_set_nat_nat(gcd_Gcd_nat,A3)),A3)) ) ) ).

% Gcd_in
tff(fact_9828_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ) ) ).

% gcd_non_0_nat
tff(fact_9829_gcd__nat_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X ) )
      & ( ( Y != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ) ) ) ).

% gcd_nat.simps
tff(fact_9830_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa2) = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y = X ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)) ) ) ) ) ).

% gcd_nat.elims
tff(fact_9831_gcd__le2__nat,axiom,
    ! [B3: nat,A2: nat] :
      ( ( B3 != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),B3)),B3)) ) ).

% gcd_le2_nat
tff(fact_9832_gcd__le1__nat,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),B3)),A2)) ) ).

% gcd_le1_nat
tff(fact_9833_gcd__diff1__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).

% gcd_diff1_nat
tff(fact_9834_gcd__diff2__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).

% gcd_diff2_nat
tff(fact_9835_bezout__nat,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 != zero_zero_nat )
     => ? [X5: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),B3)) ) ).

% bezout_nat
tff(fact_9836_bezout__gcd__nat_H,axiom,
    ! [B3: nat,A2: nat] :
    ? [X5: nat,Y3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X5)))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X5)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),B3) ) )
      | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),X5)))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B3),X5)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),B3) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9837_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] : aa_set_nat_nat(gcd_Gcd_nat,set_nat2(Xs)) = aa_nat_nat(fold_nat_nat(gcd_gcd_nat,Xs),zero_zero_nat) ).

% Gcd_nat_set_eq_fold
tff(fact_9838_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_wv) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9839_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ww(N),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9840_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa2) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = X ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).

% gcd_nat.pelims
tff(fact_9841_Field__natLeq__on,axiom,
    ! [N: nat] : field_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_wx(N)))) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bk,N)) ).

% Field_natLeq_on
tff(fact_9842_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)) ).

% natLess_def
tff(fact_9843_wf__less,axiom,
    pp(aa_set3135850010578026034t_bool(wf_nat,collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)))) ).

% wf_less
tff(fact_9844_cauchyD,axiom,
    ! [X6: fun_nat_rat,R2: rat] :
      ( cauchy(X6)
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
       => ? [K2: nat] :
          ! [M2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M2))
           => ! [N5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(X6,M2)),aa_nat_rat(X6,N5)))),R2)) ) ) ) ) ).

% cauchyD
tff(fact_9845_cauchy__imp__bounded,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ? [B: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
          & ! [N5: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N5))),B)) ) ) ).

% cauchy_imp_bounded
tff(fact_9846_cauchy__def,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
    <=> ! [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
         => ? [K3: nat] :
            ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),M3))
             => ! [N3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(X6,M3)),aa_nat_rat(X6,N3)))),R5)) ) ) ) ) ).

% cauchy_def
tff(fact_9847_cauchyI,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [R3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R3))
         => ? [K6: nat] :
            ! [M4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),M4))
             => ! [N2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(X6,M4)),aa_nat_rat(X6,N2)))),R3)) ) ) )
     => cauchy(X6) ) ).

% cauchyI
tff(fact_9848_le__Real,axiom,
    ! [X6: fun_nat_rat,Y5: fun_nat_rat] :
      ( cauchy(X6)
     => ( cauchy(Y5)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,real2(X6)),real2(Y5)))
        <=> ! [R5: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
             => ? [K3: nat] :
                ! [N3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,N3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Y5,N3)),R5))) ) ) ) ) ) ).

% le_Real
tff(fact_9849_cauchy__not__vanishes,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => ? [B: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
            & ? [K2: nat] :
              ! [N5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N5)))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_9850_vanishes__mult__bounded,axiom,
    ! [X6: fun_nat_rat,Y5: fun_nat_rat] :
      ( ? [A9: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A9))
          & ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N2))),A9)) )
     => ( vanishes(Y5)
       => vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_wy(X6),Y5)) ) ) ).

% vanishes_mult_bounded
tff(fact_9851_vanishes__def,axiom,
    ! [X6: fun_nat_rat] :
      ( vanishes(X6)
    <=> ! [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
         => ? [K3: nat] :
            ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N3))),R5)) ) ) ) ).

% vanishes_def
tff(fact_9852_vanishesI,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [R3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R3))
         => ? [K6: nat] :
            ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N2))),R3)) ) )
     => vanishes(X6) ) ).

% vanishesI
tff(fact_9853_vanishesD,axiom,
    ! [X6: fun_nat_rat,R2: rat] :
      ( vanishes(X6)
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
       => ? [K2: nat] :
          ! [N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N5))),R2)) ) ) ) ).

% vanishesD
tff(fact_9854_cauchy__not__vanishes__cases,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => ? [B: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
            & ? [K2: nat] :
                ( ! [N5: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(X6,N5)))) )
                | ! [N5: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_nat_rat(X6,N5))) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_9855_not__positive__Real,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( ~ pp(aa_real_bool(positive2,real2(X6)))
      <=> ! [R5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
           => ? [K3: nat] :
              ! [N3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,N3)),R5)) ) ) ) ) ).

% not_positive_Real
tff(fact_9856_positive__Real,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( pp(aa_real_bool(positive2,real2(X6)))
      <=> ? [R5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
            & ? [K3: nat] :
              ! [N3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(X6,N3))) ) ) ) ) ).

% positive_Real
tff(fact_9857_less__real__def,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> pp(aa_real_bool(positive2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),X))) ) ).

% less_real_def
tff(fact_9858_Real_Opositive_Orep__eq,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(positive2,X))
    <=> ? [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
          & ? [K3: nat] :
            ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(aa_real_fun_nat_rat(rep_real,X),N3))) ) ) ) ).

% Real.positive.rep_eq
tff(fact_9859_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% mask_nat_positive_iff
tff(fact_9860_not__mask__negative__int,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),zero_zero_int)) ).

% not_mask_negative_int
tff(fact_9861_mask__nonnegative__int,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,N))) ).

% mask_nonnegative_int
tff(fact_9862_less__eq__mask,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ).

% less_eq_mask
tff(fact_9863_less__mask,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ) ).

% less_mask
tff(fact_9864_mask__nat__less__exp,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% mask_nat_less_exp
tff(fact_9865_rat__less__eq__code,axiom,
    ! [P3: rat,Q3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,P3),Q3))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_xa(Q3)),quotient_of(P3))) ) ).

% rat_less_eq_code
tff(fact_9866_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q3: int] :
      ( ( quotient_of(R2) = product_Pair_int_int(P3,Q3) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q3)) ) ).

% quotient_of_denom_pos
tff(fact_9867_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(quotient_of(R2)))) ).

% quotient_of_denom_pos'
tff(fact_9868_rat__less__code,axiom,
    ! [P3: rat,Q3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P3),Q3))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_xc(Q3)),quotient_of(P3))) ) ).

% rat_less_code
tff(fact_9869_quotient__of__def,axiom,
    ! [X: rat] : quotient_of(X) = the_Pr4378521158711661632nt_int(aTP_Lamm_xd(X)) ).

% quotient_of_def
tff(fact_9870_of__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,X)) = aa_complex_complex(csqrt,aa_real_complex(real_V4546457046886955230omplex,X)) ) ) ).

% of_real_sqrt
tff(fact_9871_normalize__stable,axiom,
    ! [Q3: int,P3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,P3),Q3))
       => ( normalize(product_Pair_int_int(P3,Q3)) = product_Pair_int_int(P3,Q3) ) ) ) ).

% normalize_stable
tff(fact_9872_Rat__induct,axiom,
    ! [P: fun_rat_bool,Q3: rat] :
      ( ! [A: int,B: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A),B))
           => pp(aa_rat_bool(P,aa_int_rat(fract(A),B))) ) )
     => pp(aa_rat_bool(P,Q3)) ) ).

% Rat_induct
tff(fact_9873_Rat__cases,axiom,
    ! [Q3: rat] :
      ~ ! [A: int,B: int] :
          ( ( Q3 = aa_int_rat(fract(A),B) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A),B)) ) ) ).

% Rat_cases
tff(fact_9874_Rat__cases__nonzero,axiom,
    ! [Q3: rat] :
      ( ! [A: int,B: int] :
          ( ( Q3 = aa_int_rat(fract(A),B) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
           => ( ( A != zero_zero_int )
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A),B)) ) ) )
     => ( Q3 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
tff(fact_9875_quotient__of__unique,axiom,
    ! [R2: rat] :
    ? [X5: product_prod_int_int] :
      ( ( R2 = aa_int_rat(fract(product_fst_int_int(X5)),product_snd_int_int(X5)) )
      & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(X5)))
      & pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(X5)),product_snd_int_int(X5)))
      & ! [Y4: product_prod_int_int] :
          ( ( ( R2 = aa_int_rat(fract(product_fst_int_int(Y4)),product_snd_int_int(Y4)) )
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(Y4)))
            & pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(Y4)),product_snd_int_int(Y4))) )
         => ( Y4 = X5 ) ) ) ).

% quotient_of_unique
tff(fact_9876_coprime__Suc__0__right,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(suc,zero_zero_nat))) ).

% coprime_Suc_0_right
tff(fact_9877_coprime__Suc__0__left,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ).

% coprime_Suc_0_left
tff(fact_9878_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),N)) ) ).

% coprime_diff_one_left_nat
tff(fact_9879_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% coprime_diff_one_right_nat
tff(fact_9880_Rats__abs__nat__div__natE,axiom,
    ! [X: real] :
      ( pp(aa_set_real_bool(member_real(X),field_5140801741446780682s_real))
     => ~ ! [M4: nat,N2: nat] :
            ( ( N2 != zero_zero_nat )
           => ( ( aa_real_real(abs_abs_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,M4)),aa_nat_real(semiri5074537144036343181t_real,N2)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,M4),N2)) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_9881_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
       => ( set_or4665077453230672383an_nat(M,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),set_or4665077453230672383an_nat(M,aa_num_nat(pred_numeral,K))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
       => ( set_or4665077453230672383an_nat(M,aa_num_nat(numeral_numeral_nat,K)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_9882_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_9883_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(pred_numeral,K))) ) ).

% less_Suc_numeral
tff(fact_9884_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(pred_numeral,K)),N)) ) ).

% less_numeral_Suc
tff(fact_9885_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),N)) ) ).

% le_numeral_Suc
tff(fact_9886_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_num_nat(pred_numeral,K))) ) ).

% le_Suc_numeral
tff(fact_9887_lessThan__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(pred_numeral,K))) ).

% lessThan_nat_numeral
tff(fact_9888_atMost__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(numeral_numeral_nat,K),aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(pred_numeral,K))) ).

% atMost_nat_numeral
tff(fact_9889_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( last_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_9890_Real_Opositive_Oabs__eq,axiom,
    ! [X: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
     => ( pp(aa_real_bool(positive2,real2(X)))
      <=> ? [R5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
            & ? [K3: nat] :
              ! [N3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(X,N3))) ) ) ) ) ).

% Real.positive.abs_eq
tff(fact_9891_not__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% not_negative_int_iff
tff(fact_9892_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% not_nonnegative_int_iff
tff(fact_9893_Real_Opositive_Orsp,axiom,
    bNF_re2717234250851183977l_bool(realrel,fequal_bool,aTP_Lamm_xe,aTP_Lamm_xe) ).

% Real.positive.rsp
tff(fact_9894_less__eq__integer_Orsp,axiom,
    bNF_re5780769757191362231t_bool(fequal_int,bNF_re9007465950226028919l_bool(fequal_int,fequal_bool),ord_less_eq_int,ord_less_eq_int) ).

% less_eq_integer.rsp
tff(fact_9895_less__eq__natural_Orsp,axiom,
    bNF_re1480548845665215743t_bool(fequal_nat,bNF_re6223507645442431167l_bool(fequal_nat,fequal_bool),ord_less_eq_nat,ord_less_eq_nat) ).

% less_eq_natural.rsp
tff(fact_9896_less__natural_Orsp,axiom,
    bNF_re1480548845665215743t_bool(fequal_nat,bNF_re6223507645442431167l_bool(fequal_nat,fequal_bool),ord_less_nat,ord_less_nat) ).

% less_natural.rsp
tff(fact_9897_less__integer_Orsp,axiom,
    bNF_re5780769757191362231t_bool(fequal_int,bNF_re9007465950226028919l_bool(fequal_int,fequal_bool),ord_less_int,ord_less_int) ).

% less_integer.rsp
tff(fact_9898_Real_Opositive_Otransfer,axiom,
    bNF_re1458715823057739248l_bool(pcr_real,fequal_bool,aTP_Lamm_xe,positive2) ).

% Real.positive.transfer
tff(fact_9899_Rat_Opositive_Otransfer,axiom,
    bNF_re3535474257078034868l_bool(pcr_rat,fequal_bool,aTP_Lamm_xf,positive) ).

% Rat.positive.transfer
tff(fact_9900_less__eq__int_Otransfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_wn),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_9901_zero__int_Otransfer,axiom,
    pp(aa_int_bool(aa_Pro2928840338864870335t_bool(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),zero_zero_int)) ).

% zero_int.transfer
tff(fact_9902_int__transfer,axiom,
    bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_xg,semiri1314217659103216013at_int) ).

% int_transfer
tff(fact_9903_one__int_Otransfer,axiom,
    pp(aa_int_bool(aa_Pro2928840338864870335t_bool(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),one_one_int)) ).

% one_int.transfer
tff(fact_9904_less__int_Otransfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_wp),ord_less_int) ).

% less_int.transfer
tff(fact_9905_Rat_Opositive_Orsp,axiom,
    bNF_re1394333895916556125l_bool(ratrel,fequal_bool,aTP_Lamm_xf,aTP_Lamm_xf) ).

% Rat.positive.rsp
tff(fact_9906_Rat_Opositive_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
     => ( pp(aa_rat_bool(positive,abs_Rat(X)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X)),product_snd_int_int(X)))) ) ) ).

% Rat.positive.abs_eq
tff(fact_9907_vimage__Suc__insert__0,axiom,
    ! [A3: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A3)) = vimage_nat_nat(suc,A3) ).

% vimage_Suc_insert_0
tff(fact_9908_finite__vimage__Suc__iff,axiom,
    ! [F2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,vimage_nat_nat(suc,F2)))
    <=> pp(aa_set_nat_bool(finite_finite_nat,F2)) ) ).

% finite_vimage_Suc_iff
tff(fact_9909_pairs__le__eq__Sigma,axiom,
    ! [M: nat] : collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_xh(M))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,M),aTP_Lamm_xi(M)) ).

% pairs_le_eq_Sigma
tff(fact_9910_Restr__natLeq,axiom,
    ! [N: nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bk,N)),aTP_Lamm_xj(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_wx(N))) ).

% Restr_natLeq
tff(fact_9911_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_9912_Restr__natLeq2,axiom,
    ! [N: nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq,N),aTP_Lamm_xk(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_wx(N))) ).

% Restr_natLeq2
tff(fact_9913_natLeq__underS__less,axiom,
    ! [N: nat] : order_underS_nat(bNF_Ca8665028551170535155natLeq,N) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bk,N)) ).

% natLeq_underS_less
tff(fact_9914_natLeq__on__wo__rel,axiom,
    ! [N: nat] : pp(aa_set3135850010578026034t_bool(bNF_We3818239936649020644el_nat,collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_wx(N))))) ).

% natLeq_on_wo_rel
tff(fact_9915_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N)),transi2905341329935302413cl_nat(pred_nat)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% pred_nat_trancl_eq_le
tff(fact_9916_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F: fun_nat_real,M5: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_hz(F),M5),at_top_nat)
     => ( ! [M4: nat,N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),M4))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,M4)),aa_nat_real(F,N2))) ) )
       => topolo7531315842566124627t_real(F) ) ) ).

% Bseq_monoseq_convergent'_inc
tff(fact_9917_Bseq__mono__convergent,axiom,
    ! [X6: fun_nat_real] :
      ( bfun_nat_real(X6,at_top_nat)
     => ( ! [M4: nat,N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,M4)),aa_nat_real(X6,N2))) )
       => topolo7531315842566124627t_real(X6) ) ) ).

% Bseq_mono_convergent
tff(fact_9918_convergent__realpow,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => topolo7531315842566124627t_real(aa_real_fun_nat_real(power_power_real,X)) ) ) ).

% convergent_realpow
tff(fact_9919_less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N)),transi6264000038957366511cl_nat(pred_nat)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% less_eq
tff(fact_9920_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F: fun_nat_real,M5: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_hz(F),M5),at_top_nat)
     => ( ! [M4: nat,N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),M4))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,M4))) ) )
       => topolo7531315842566124627t_real(F) ) ) ).

% Bseq_monoseq_convergent'_dec
tff(fact_9921_pair__lessI2,axiom,
    ! [A2: nat,B3: nat,S2: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S2),T))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),T)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_9922_trans__pair__less,axiom,
    trans_4347625901269045472at_nat(fun_pair_less) ).

% trans_pair_less
tff(fact_9923_total__pair__less,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : total_3592101749530773125at_nat(A3,fun_pair_less) ).

% total_pair_less
tff(fact_9924_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z)),fun_pair_less)
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z)) ) ).

% pair_less_iff1
tff(fact_9925_wf__pair__less,axiom,
    wf_Pro7803398752247294826at_nat(fun_pair_less) ).

% wf_pair_less
tff(fact_9926_pair__lessI1,axiom,
    ! [A2: nat,B3: nat,S2: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),T)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_9927_gcd__nat_Oordering__top__axioms,axiom,
    pp(aa_nat_bool(ordering_top_nat(dvd_dvd_nat,aTP_Lamm_wv),zero_zero_nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_9928_bot__nat__0_Oordering__top__axioms,axiom,
    pp(aa_nat_bool(ordering_top_nat(aTP_Lamm_bl,aTP_Lamm_bk),zero_zero_nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_9929_pair__leqI2,axiom,
    ! [A2: nat,B3: nat,S2: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,S2),T))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),T)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_9930_pair__leqI1,axiom,
    ! [A2: nat,B3: nat,S2: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),T)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_9931_pair__leq__def,axiom,
    fun_pair_leq = aa_set5535132180132642428at_nat(aa_set3125807151510273166at_nat(sup_su718114333110466843at_nat,fun_pair_less),id_Pro2258643101195443293at_nat) ).

% pair_leq_def
tff(fact_9932_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),XS))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_min_weak)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS)),fun_min_weak) ) ) ) ).

% wmin_insertI
tff(fact_9933_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),YS))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_max_weak)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,XS),YS),fun_max_weak) ) ) ) ).

% wmax_insertI
tff(fact_9934_wmin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] : member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,bot_bo2099793752762293965at_nat),fun_min_weak) ).

% wmin_emptyI
tff(fact_9935_wmax__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,X6))
     => member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,X6),fun_max_weak) ) ).

% wmax_emptyI
tff(fact_9936_max__weak__def,axiom,
    fun_max_weak = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,max_ex8135407076693332796at_nat(fun_pair_leq)),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% max_weak_def
tff(fact_9937_min__weak__def,axiom,
    fun_min_weak = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,min_ex6901939911449802026at_nat(fun_pair_leq)),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% min_weak_def
tff(fact_9938_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),XS))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_min_strict)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS)),fun_min_strict) ) ) ) ).

% smin_insertI
tff(fact_9939_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y5: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X6: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),Y5))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,Y5),fun_max_strict)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,X6),Y5),fun_max_strict) ) ) ) ).

% smax_insertI
tff(fact_9940_smax__emptyI,axiom,
    ! [Y5: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,Y5))
     => ( ( Y5 != bot_bo2099793752762293965at_nat )
       => member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,Y5),fun_max_strict) ) ) ).

% smax_emptyI
tff(fact_9941_smin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( ( X6 != bot_bo2099793752762293965at_nat )
     => member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,bot_bo2099793752762293965at_nat),fun_min_strict) ) ).

% smin_emptyI
tff(fact_9942_min__strict__def,axiom,
    fun_min_strict = min_ex6901939911449802026at_nat(fun_pair_less) ).

% min_strict_def
tff(fact_9943_max__strict__def,axiom,
    fun_max_strict = max_ex8135407076693332796at_nat(fun_pair_less) ).

% max_strict_def
tff(fact_9944_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_min_strict,fun_min_weak)) ).

% min_rpair_set
tff(fact_9945_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_max_strict,fun_max_weak)) ).

% max_rpair_set
tff(fact_9946_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),K),L)),N))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
          & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_9947_concat__bit__0,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(zero_zero_nat),K),L) = L ).

% concat_bit_0
tff(fact_9948_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ).

% concat_bit_nonnegative_iff
tff(fact_9949_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ).

% concat_bit_negative_iff
tff(fact_9950_less__eq__int_Orsp,axiom,
    bNF_re2219331435504962769t_bool(intrel,bNF_re3189209160384574813l_bool(intrel,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_wn),produc5023667602222594023t_bool(aTP_Lamm_wn)) ).

% less_eq_int.rsp
tff(fact_9951_zero__int_Orsp,axiom,
    pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat))) ).

% zero_int.rsp
tff(fact_9952_one__int_Orsp,axiom,
    pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat))) ).

% one_int.rsp
tff(fact_9953_less__int_Orsp,axiom,
    bNF_re2219331435504962769t_bool(intrel,bNF_re3189209160384574813l_bool(intrel,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_wp),produc5023667602222594023t_bool(aTP_Lamm_wp)) ).

% less_int.rsp
tff(fact_9954_division__segment__int__def,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
       => ( aa_int_int(euclid3395696857347342551nt_int,K) = one_one_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
       => ( aa_int_int(euclid3395696857347342551nt_int,K) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% division_segment_int_def
tff(fact_9955_less__eq__enat__def,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,M),N))
    <=> pp(aa_Ext2772979874139974381t_bool(aa_boo1364755444788961487t_bool(extend8126323780198025762t_bool(aa_Ext6309394871002001526t_bool(aTP_Lamm_xl,M)),fTrue),N)) ) ).

% less_eq_enat_def
tff(fact_9956_less__enat__def,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,M),N))
    <=> pp(aa_Ext2772979874139974381t_bool(aa_boo1364755444788961487t_bool(extend8126323780198025762t_bool(aa_Ext6309394871002001526t_bool(aTP_Lamm_xm,N)),fFalse),M)) ) ).

% less_enat_def
tff(fact_9957_of__rat__dense,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [Q4: rat] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_rat_real(field_7254667332652039916t_real,Q4)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(field_7254667332652039916t_real,Q4)),Y)) ) ) ).

% of_rat_dense
tff(fact_9958_less__RealD,axiom,
    ! [Y5: fun_nat_rat,X: real] :
      ( cauchy(Y5)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),real2(Y5)))
       => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(Y5,N2)))) ) ) ).

% less_RealD
tff(fact_9959_Real__leI,axiom,
    ! [X6: fun_nat_rat,Y: real] :
      ( cauchy(X6)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(X6,N2))),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,real2(X6)),Y)) ) ) ).

% Real_leI
tff(fact_9960_le__RealI,axiom,
    ! [Y5: fun_nat_rat,X: real] :
      ( cauchy(Y5)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(Y5,N2))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),real2(Y5))) ) ) ).

% le_RealI
tff(fact_9961_compute__powr__real,axiom,
    ! [B3: real,I: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),zero_zero_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real2,B3),I) = abort_real(literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fFalse,fTrue,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,zero_zero_literal))))))))))))))))))))))))))))))),aa_rea7683167438103758690t_real(aTP_Lamm_xn(B3),I)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),zero_zero_real))
       => ( ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,I)) = I )
           => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),I))
               => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real2,B3),I) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,I))) ) )
              & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),I))
               => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real2,B3),I) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,B3),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,I))))) ) ) ) )
          & ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,I)) != I )
           => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real2,B3),I) = abort_real(literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fTrue,fFalse,fTrue,fFalse,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,zero_zero_literal))))))))))))))))))))))))))))))))))),aa_rea7683167438103758690t_real(aTP_Lamm_xn(B3),I)) ) ) ) ) ) ).

% compute_powr_real
tff(fact_9962_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,set_or4665077453230672383an_nat(zero_zero_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% UNIV_char_of_nat
tff(fact_9963_inj__on__char__of__nat,axiom,
    pp(aa_set_nat_bool(inj_on_nat_char(unique3096191561947761185of_nat),set_or4665077453230672383an_nat(zero_zero_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))))) ).

% inj_on_char_of_nat
tff(fact_9964_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = set_or4665077453230672383an_nat(zero_zero_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% range_nat_of_char
tff(fact_9965_nat__of__char__less__256,axiom,
    ! [C: char] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% nat_of_char_less_256
tff(fact_9966_numeral__le__enat__iff,axiom,
    ! [M: num,N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(M)),extended_enat2(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),N)) ) ).

% numeral_le_enat_iff
tff(fact_9967_enat__ord__simps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,extended_enat2(M)),extended_enat2(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% enat_ord_simps(2)
tff(fact_9968_enat__ord__simps_I1_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,extended_enat2(M)),extended_enat2(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% enat_ord_simps(1)
tff(fact_9969_idiff__enat__0__right,axiom,
    ! [N: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(N),extended_enat2(zero_zero_nat)) = N ).

% idiff_enat_0_right
tff(fact_9970_idiff__enat__0,axiom,
    ! [N: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(extended_enat2(zero_zero_nat)),N) = extended_enat2(zero_zero_nat) ).

% idiff_enat_0
tff(fact_9971_numeral__less__enat__iff,axiom,
    ! [M: num,N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(M)),extended_enat2(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),N)) ) ).

% numeral_less_enat_iff
tff(fact_9972_zero__enat__def,axiom,
    zero_z5237406670263579293d_enat = extended_enat2(zero_zero_nat) ).

% zero_enat_def
tff(fact_9973_enat__0__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( extended_enat2(X) = zero_z5237406670263579293d_enat )
    <=> ( X = zero_zero_nat ) ) ).

% enat_0_iff(1)
tff(fact_9974_enat__0__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( zero_z5237406670263579293d_enat = extended_enat2(X) )
    <=> ( X = zero_zero_nat ) ) ).

% enat_0_iff(2)
tff(fact_9975_finite__enat__bounded,axiom,
    ! [A3: set_Extended_enat,N: nat] :
      ( ! [Y3: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y3),A3))
         => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,Y3),extended_enat2(N))) )
     => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3)) ) ).

% finite_enat_bounded
tff(fact_9976_enat__ile,axiom,
    ! [N: extended_enat,M: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,N),extended_enat2(M)))
     => ? [K2: nat] : N = extended_enat2(K2) ) ).

% enat_ile
tff(fact_9977_enat__iless,axiom,
    ! [N: extended_enat,M: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,N),extended_enat2(M)))
     => ? [K2: nat] : N = extended_enat2(K2) ) ).

% enat_iless
tff(fact_9978_less__enatE,axiom,
    ! [N: extended_enat,M: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,N),extended_enat2(M)))
     => ~ ! [K2: nat] :
            ( ( N = extended_enat2(K2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),M)) ) ) ).

% less_enatE
tff(fact_9979_Suc__ile__eq,axiom,
    ! [M: nat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,extended_enat2(aa_nat_nat(suc,M))),N))
    <=> pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,extended_enat2(M)),N)) ) ).

% Suc_ile_eq
tff(fact_9980_iadd__le__enat__iff,axiom,
    ! [X: extended_enat,Y: extended_enat,N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),Y)),extended_enat2(N)))
    <=> ? [Y6: nat,X8: nat] :
          ( ( X = extended_enat2(X8) )
          & ( Y = extended_enat2(Y6) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X8),Y6)),N)) ) ) ).

% iadd_le_enat_iff
tff(fact_9981_elimnum,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary)),N))
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info2,Deg,TreeList2,Summary),extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = vEBT_Node(Info2,Deg,TreeList2,Summary) ) ) ).

% elimnum
tff(fact_9982_times__enat__simps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extend5688581933313929465d_enat),extended_enat2(N)) = zero_z5237406670263579293d_enat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extend5688581933313929465d_enat),extended_enat2(N)) = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(3)
tff(fact_9983_elimcomplete,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary)),N))
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info2,Deg,TreeList2,Summary),extend5688581933313929465d_enat) = vEBT_Node(Info2,Deg,TreeList2,Summary) ) ) ).

% elimcomplete
tff(fact_9984_enat__ord__simps_I4_J,axiom,
    ! [Q3: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,Q3),extend5688581933313929465d_enat))
    <=> ( Q3 != extend5688581933313929465d_enat ) ) ).

% enat_ord_simps(4)
tff(fact_9985_enat__ord__simps_I6_J,axiom,
    ! [Q3: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,extend5688581933313929465d_enat),Q3)) ).

% enat_ord_simps(6)
tff(fact_9986_enat__ord__code_I3_J,axiom,
    ! [Q3: extended_enat] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,Q3),extend5688581933313929465d_enat)) ).

% enat_ord_code(3)
tff(fact_9987_enat__ord__simps_I5_J,axiom,
    ! [Q3: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,extend5688581933313929465d_enat),Q3))
    <=> ( Q3 = extend5688581933313929465d_enat ) ) ).

% enat_ord_simps(5)
tff(fact_9988_times__enat__simps_I4_J,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extended_enat2(M)),extend5688581933313929465d_enat) = zero_z5237406670263579293d_enat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extended_enat2(M)),extend5688581933313929465d_enat) = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(4)
tff(fact_9989_enat__ord__code_I5_J,axiom,
    ! [N: nat] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,extend5688581933313929465d_enat),extended_enat2(N))) ).

% enat_ord_code(5)
tff(fact_9990_infinity__ileE,axiom,
    ! [M: nat] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,extend5688581933313929465d_enat),extended_enat2(M))) ).

% infinity_ileE
tff(fact_9991_enat__add__left__cancel__less,axiom,
    ! [A2: extended_enat,B3: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A2),B3)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A2),C)))
    <=> ( ( A2 != extend5688581933313929465d_enat )
        & pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,B3),C)) ) ) ).

% enat_add_left_cancel_less
tff(fact_9992_enat__ord__code_I4_J,axiom,
    ! [M: nat] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,extended_enat2(M)),extend5688581933313929465d_enat)) ).

% enat_ord_code(4)
tff(fact_9993_less__infinityE,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,N),extend5688581933313929465d_enat))
     => ~ ! [K2: nat] : N != extended_enat2(K2) ) ).

% less_infinityE
tff(fact_9994_infinity__ilessE,axiom,
    ! [M: nat] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,extend5688581933313929465d_enat),extended_enat2(M))) ).

% infinity_ilessE
tff(fact_9995_enat__add__left__cancel__le,axiom,
    ! [A2: extended_enat,B3: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A2),B3)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A2),C)))
    <=> ( ( A2 = extend5688581933313929465d_enat )
        | pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,B3),C)) ) ) ).

% enat_add_left_cancel_le
tff(fact_9996_enat__ord__simps_I3_J,axiom,
    ! [Q3: extended_enat] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,Q3),extend5688581933313929465d_enat)) ).

% enat_ord_simps(3)
tff(fact_9997_Inf__enat__def,axiom,
    ! [A3: set_Extended_enat] :
      ( ( ( A3 = bot_bo7653980558646680370d_enat )
       => ( comple2295165028678016749d_enat(A3) = extend5688581933313929465d_enat ) )
      & ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( comple2295165028678016749d_enat(A3) = ord_Le1955565732374568822d_enat(aTP_Lamm_xo(A3)) ) ) ) ).

% Inf_enat_def
tff(fact_9998_Sup__enat__def,axiom,
    ! [A3: set_Extended_enat] :
      ( ( ( A3 = bot_bo7653980558646680370d_enat )
       => ( comple4398354569131411667d_enat(A3) = zero_z5237406670263579293d_enat ) )
      & ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
           => ( comple4398354569131411667d_enat(A3) = lattic921264341876707157d_enat(A3) ) )
          & ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
           => ( comple4398354569131411667d_enat(A3) = extend5688581933313929465d_enat ) ) ) ) ) ).

% Sup_enat_def
tff(fact_9999_imult__infinity,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),N))
     => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extend5688581933313929465d_enat),N) = extend5688581933313929465d_enat ) ) ).

% imult_infinity
tff(fact_10000_imult__infinity__right,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),N))
     => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(N),extend5688581933313929465d_enat) = extend5688581933313929465d_enat ) ) ).

% imult_infinity_right
tff(fact_10001_times__enat__def,axiom,
    ! [M: extended_enat,N: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(M),N) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_xq(N),aa_Ext7969882492699152700d_enat(if_Extended_enat(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,N),zero_z5237406670263579293d_enat),zero_z5237406670263579293d_enat),extend5688581933313929465d_enat)),M) ).

% times_enat_def
tff(fact_10002_iless__Suc__eq,axiom,
    ! [M: nat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,extended_enat2(M)),aa_Ext7969882492699152700d_enat(extended_eSuc,N)))
    <=> pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,extended_enat2(M)),N)) ) ).

% iless_Suc_eq
tff(fact_10003_eSuc__mono,axiom,
    ! [N: extended_enat,M: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(extended_eSuc,N)),aa_Ext7969882492699152700d_enat(extended_eSuc,M)))
    <=> pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,N),M)) ) ).

% eSuc_mono
tff(fact_10004_eSuc__ile__mono,axiom,
    ! [N: extended_enat,M: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(extended_eSuc,N)),aa_Ext7969882492699152700d_enat(extended_eSuc,M)))
    <=> pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,N),M)) ) ).

% eSuc_ile_mono
tff(fact_10005_iless__eSuc0,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,N),aa_Ext7969882492699152700d_enat(extended_eSuc,zero_z5237406670263579293d_enat)))
    <=> ( N = zero_z5237406670263579293d_enat ) ) ).

% iless_eSuc0
tff(fact_10006_eSuc__Max,axiom,
    ! [A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( aa_Ext7969882492699152700d_enat(extended_eSuc,lattic921264341876707157d_enat(A3)) = lattic921264341876707157d_enat(image_80655429650038917d_enat(extended_eSuc,A3)) ) ) ) ).

% eSuc_Max
tff(fact_10007_eSuc__Sup,axiom,
    ! [A3: set_Extended_enat] :
      ( ( A3 != bot_bo7653980558646680370d_enat )
     => ( aa_Ext7969882492699152700d_enat(extended_eSuc,comple4398354569131411667d_enat(A3)) = comple4398354569131411667d_enat(image_80655429650038917d_enat(extended_eSuc,A3)) ) ) ).

% eSuc_Sup
tff(fact_10008_not__eSuc__ilei0,axiom,
    ! [N: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(extended_eSuc,N)),zero_z5237406670263579293d_enat)) ).

% not_eSuc_ilei0
tff(fact_10009_ile__eSuc,axiom,
    ! [N: extended_enat] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,N),aa_Ext7969882492699152700d_enat(extended_eSuc,N))) ).

% ile_eSuc
tff(fact_10010_ileI1,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,M),N))
     => pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(extended_eSuc,M)),N)) ) ).

% ileI1
tff(fact_10011_i0__iless__eSuc,axiom,
    ! [N: extended_enat] : pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(extended_eSuc,N))) ).

% i0_iless_eSuc
tff(fact_10012_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),less_than))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% less_than_iff
tff(fact_10013_pair__less__def,axiom,
    fun_pair_less = lex_prod_nat_nat(less_than,less_than) ).

% pair_less_def
tff(fact_10014_natLeq__on__well__order__on,axiom,
    ! [N: nat] : pp(aa_set3135850010578026034t_bool(order_2888998067076097458on_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bk,N))),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_wx(N))))) ).

% natLeq_on_well_order_on
tff(fact_10015_natLeq__on__Well__order,axiom,
    ! [N: nat] : pp(aa_set3135850010578026034t_bool(order_2888998067076097458on_nat(field_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_wx(N))))),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_wx(N))))) ).

% natLeq_on_Well_order
tff(fact_10016_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aTP_Lamm_xr),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_10017_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fTrue,aTP_Lamm_et),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_10018_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aa_nat_fun_nat_bool(ord_less_eq_nat,M)),N)) ) ).

% less_eq_nat.simps(2)
tff(fact_10019_max__Suc1,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_xs,N)),M) ).

% max_Suc1
tff(fact_10020_max__Suc2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_xt,N)),M) ).

% max_Suc2
tff(fact_10021_diff__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_qb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ).

% diff_Suc
tff(fact_10022_min__Suc2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_xu,N)),M) ).

% min_Suc2
tff(fact_10023_min__Suc1,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_xv,N)),M) ).

% min_Suc1
tff(fact_10024_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_qb),Nat) ).

% pred_def
tff(fact_10025_UNIV__bool,axiom,
    top_top_set_bool = insert_bool(fFalse,insert_bool(fTrue,bot_bot_set_bool)) ).

% UNIV_bool
tff(fact_10026_Rep__unit__induct,axiom,
    ! [Y: bool,P: fun_bool_bool] :
      ( pp(aa_set_bool_bool(member_bool(Y),insert_bool(fTrue,bot_bot_set_bool)))
     => ( ! [X5: product_unit] : pp(aa_bool_bool(P,aa_Product_unit_bool(product_Rep_unit,X5)))
       => pp(aa_bool_bool(P,Y)) ) ) ).

% Rep_unit_induct
tff(fact_10027_Abs__unit__inject,axiom,
    ! [X: bool,Y: bool] :
      ( pp(aa_set_bool_bool(member_bool(X),insert_bool(fTrue,bot_bot_set_bool)))
     => ( pp(aa_set_bool_bool(member_bool(Y),insert_bool(fTrue,bot_bot_set_bool)))
       => ( ( aa_bool_Product_unit(product_Abs_unit,X) = aa_bool_Product_unit(product_Abs_unit,Y) )
        <=> ( pp(X)
          <=> pp(Y) ) ) ) ) ).

% Abs_unit_inject
tff(fact_10028_Abs__unit__inverse,axiom,
    ! [Y: bool] :
      ( pp(aa_set_bool_bool(member_bool(Y),insert_bool(fTrue,bot_bot_set_bool)))
     => ( pp(aa_Product_unit_bool(product_Rep_unit,aa_bool_Product_unit(product_Abs_unit,Y)))
      <=> pp(Y) ) ) ).

% Abs_unit_inverse
tff(fact_10029_Rep__unit,axiom,
    ! [X: product_unit] : pp(aa_set_bool_bool(member_bool(aa_Product_unit_bool(product_Rep_unit,X)),insert_bool(fTrue,bot_bot_set_bool))) ).

% Rep_unit
tff(fact_10030_Abs__unit__cases,axiom,
    ! [X: product_unit] :
      ~ ! [Y3: bool] :
          ( ( X = aa_bool_Product_unit(product_Abs_unit,Y3) )
         => ~ pp(aa_set_bool_bool(member_bool(Y3),insert_bool(fTrue,bot_bot_set_bool))) ) ).

% Abs_unit_cases
tff(fact_10031_Rep__unit__cases,axiom,
    ! [Y: bool] :
      ( pp(aa_set_bool_bool(member_bool(Y),insert_bool(fTrue,bot_bot_set_bool)))
     => ~ ! [X5: product_unit] :
            ( pp(Y)
          <=> ~ pp(aa_Product_unit_bool(product_Rep_unit,X5)) ) ) ).

% Rep_unit_cases
tff(fact_10032_Abs__unit__induct,axiom,
    ! [P: fun_Pr1572506220757514509t_bool,X: product_unit] :
      ( ! [Y3: bool] :
          ( pp(aa_set_bool_bool(member_bool(Y3),insert_bool(fTrue,bot_bot_set_bool)))
         => pp(aa_Product_unit_bool(P,aa_bool_Product_unit(product_Abs_unit,Y3))) )
     => pp(aa_Product_unit_bool(P,X)) ) ).

% Abs_unit_induct
tff(fact_10033_type__definition__unit,axiom,
    pp(aa_set_bool_bool(type_d5772988228442255350t_bool(product_Rep_unit,product_Abs_unit),insert_bool(fTrue,bot_bot_set_bool))) ).

% type_definition_unit
tff(fact_10034_Real_Opositive__def,axiom,
    positive2 = map_fu1722713901878115497l_bool(rep_real,id_bool,aTP_Lamm_xe) ).

% Real.positive_def
tff(fact_10035_Rat_Opositive__def,axiom,
    positive = map_fu5118516228767806217l_bool(rep_Rat,id_bool,aTP_Lamm_xf) ).

% Rat.positive_def
tff(fact_10036_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,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_10037_complex__Re__le__cmod,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,X)),aa_complex_real(real_V1022390504157884413omplex,X))) ).

% complex_Re_le_cmod
tff(fact_10038_abs__Re__le__cmod,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,X))),aa_complex_real(real_V1022390504157884413omplex,X))) ).

% abs_Re_le_cmod
tff(fact_10039_Re__csqrt,axiom,
    ! [Z: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z)))) ).

% Re_csqrt
tff(fact_10040_complex__abs__le__norm,axiom,
    ! [Z: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_complex_real(real_V1022390504157884413omplex,Z)))) ).

% complex_abs_le_norm
tff(fact_10041_csqrt__unique,axiom,
    ! [W2: complex,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z )
     => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,W2)))
          | ( ( aa_complex_real(re,W2) = zero_zero_real )
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,W2))) ) )
       => ( aa_complex_complex(csqrt,Z) = W2 ) ) ) ).

% csqrt_unique
tff(fact_10042_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(im,X) = zero_zero_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,X)))
       => ( aa_complex_complex(csqrt,X) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_complex_real(re,X))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_10043_abs__Im__le__cmod,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,X))),aa_complex_real(real_V1022390504157884413omplex,X))) ).

% abs_Im_le_cmod
tff(fact_10044_cmod__Im__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_real(re,X) = aa_complex_real(re,Y) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,X))),aa_real_real(abs_abs_real,aa_complex_real(im,Y)))) ) ) ).

% cmod_Im_le_iff
tff(fact_10045_cmod__Re__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_real(im,X) = aa_complex_real(im,Y) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,X))),aa_real_real(abs_abs_real,aa_complex_real(re,Y)))) ) ) ).

% cmod_Re_le_iff
tff(fact_10046_csqrt__principal,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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 )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(csqrt,Z)))) ) ) ).

% csqrt_principal
tff(fact_10047_cmod__le,axiom,
    ! [Z: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),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))))) ).

% cmod_le
tff(fact_10048_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% complex_neq_0
tff(fact_10049_csqrt__square,axiom,
    ! [B3: complex] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,B3)))
        | ( ( aa_complex_real(re,B3) = zero_zero_real )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,B3))) ) )
     => ( aa_complex_complex(csqrt,aa_nat_complex(power_power_complex(B3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = B3 ) ) ).

% csqrt_square
tff(fact_10050_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(im,X) = zero_zero_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,X)),zero_zero_real))
       => ( aa_complex_complex(csqrt,X) = aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_real_real(abs_abs_real,aa_complex_real(re,X))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_10051_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(im,X)),zero_zero_real))
        | ( ( aa_complex_real(im,X) = zero_zero_real )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,X))) ) )
     => ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(csqrt,X)) ) ) ).

% csqrt_minus
tff(fact_10052_complex__div__gt__0,axiom,
    ! [A2: complex,B3: complex] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A2),B3))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B3))))) )
      & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A2),B3))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B3))))) ) ) ).

% complex_div_gt_0
tff(fact_10053_Re__complex__div__gt__0,axiom,
    ! [A2: complex,B3: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A2),B3))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B3))))) ) ).

% Re_complex_div_gt_0
tff(fact_10054_Re__complex__div__lt__0,axiom,
    ! [A2: complex,B3: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A2),B3))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B3)))),zero_zero_real)) ) ).

% Re_complex_div_lt_0
tff(fact_10055_Re__complex__div__le__0,axiom,
    ! [A2: complex,B3: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A2),B3))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B3)))),zero_zero_real)) ) ).

% Re_complex_div_le_0
tff(fact_10056_Re__complex__div__ge__0,axiom,
    ! [A2: complex,B3: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A2),B3))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B3))))) ) ).

% Re_complex_div_ge_0
tff(fact_10057_Im__complex__div__gt__0,axiom,
    ! [A2: complex,B3: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A2),B3))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B3))))) ) ).

% Im_complex_div_gt_0
tff(fact_10058_Im__complex__div__lt__0,axiom,
    ! [A2: complex,B3: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A2),B3))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B3)))),zero_zero_real)) ) ).

% Im_complex_div_lt_0
tff(fact_10059_Im__complex__div__le__0,axiom,
    ! [A2: complex,B3: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A2),B3))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B3)))),zero_zero_real)) ) ).

% Im_complex_div_le_0
tff(fact_10060_Im__complex__div__ge__0,axiom,
    ! [A2: complex,B3: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A2),B3))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B3))))) ) ).

% Im_complex_div_ge_0
tff(fact_10061_less__eq__int__def,axiom,
    ord_less_eq_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_wn)) ).

% less_eq_int_def
tff(fact_10062_less__int__def,axiom,
    ord_less_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_wp)) ).

% less_int_def
tff(fact_10063_MOST__SucD,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(aTP_Lamm_uf(P),cofinite_nat)
     => eventually_nat(P,cofinite_nat) ) ).

% MOST_SucD
tff(fact_10064_MOST__SucI,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(P,cofinite_nat)
     => eventually_nat(aTP_Lamm_uf(P),cofinite_nat) ) ).

% MOST_SucI
tff(fact_10065_MOST__Suc__iff,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(aTP_Lamm_uf(P),cofinite_nat)
    <=> eventually_nat(P,cofinite_nat) ) ).

% MOST_Suc_iff
tff(fact_10066_MOST__nat__le,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(P,cofinite_nat)
    <=> ? [M3: nat] :
        ! [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N3))
         => pp(aa_nat_bool(P,N3)) ) ) ).

% MOST_nat_le
tff(fact_10067_MOST__ge__nat,axiom,
    ! [M: nat] : eventually_nat(aa_nat_fun_nat_bool(ord_less_eq_nat,M),cofinite_nat) ).

% MOST_ge_nat
tff(fact_10068_MOST__nat,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(P,cofinite_nat)
    <=> ? [M3: nat] :
        ! [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N3))
         => pp(aa_nat_bool(P,N3)) ) ) ).

% MOST_nat
tff(fact_10069_infinity__enat__def,axiom,
    extend5688581933313929465d_enat = extended_Abs_enat(none_nat) ).

% infinity_enat_def
tff(fact_10070_enat__def,axiom,
    ! [N: nat] : extended_enat2(N) = extended_Abs_enat(some_nat(N)) ).

% enat_def
tff(fact_10071_INFM__nat__le,axiom,
    ! [P: fun_nat_bool] :
      ( frequently_nat(P,cofinite_nat)
    <=> ! [M3: nat] :
        ? [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N3))
          & pp(aa_nat_bool(P,N3)) ) ) ).

% INFM_nat_le
tff(fact_10072_INFM__nat,axiom,
    ! [P: fun_nat_bool] :
      ( frequently_nat(P,cofinite_nat)
    <=> ! [M3: nat] :
        ? [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N3))
          & pp(aa_nat_bool(P,N3)) ) ) ).

% INFM_nat
tff(fact_10073_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( nat_list_encode(X) = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X5: nat,Xs2: list_nat] :
              ( ( X = cons_nat(X5,Xs2) )
             => ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),nat_list_encode(Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_10074_le__prod__encode__1,axiom,
    ! [A2: nat,B3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B3)))) ).

% le_prod_encode_1
tff(fact_10075_le__prod__encode__2,axiom,
    ! [B3: nat,A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B3)))) ).

% le_prod_encode_2
tff(fact_10076_list__encode_Osimps_I1_J,axiom,
    nat_list_encode(nil_nat) = zero_zero_nat ).

% list_encode.simps(1)
tff(fact_10077_list__encode_Opelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( nat_list_encode(X) = Y )
     => ( pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),X))
       => ( ( ( X = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),nil_nat)) ) )
         => ~ ! [X5: nat,Xs2: list_nat] :
                ( ( X = cons_nat(X5,Xs2) )
               => ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),nat_list_encode(Xs2)))) )
                 => ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),cons_nat(X5,Xs2))) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_10078_prod__decode__def,axiom,
    nat_prod_decode = nat_prod_decode_aux(zero_zero_nat) ).

% prod_decode_def
tff(fact_10079_list__decode_Opinduct,axiom,
    ! [A0: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),A0))
     => ( ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat))
         => pp(aa_nat_bool(P,zero_zero_nat)) )
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N2)))
             => ( ! [X4: nat,Y4: nat] :
                    ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y4) = aa_nat7603221163087113075at_nat(nat_prod_decode,N2) )
                   => pp(aa_nat_bool(P,Y4)) )
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,A0)) ) ) ) ).

% list_decode.pinduct
tff(fact_10080_list__decode_Oelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( nat_list_decode(X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != nil_nat ) )
       => ~ ! [N2: nat] :
              ( ( X = aa_nat_nat(suc,N2) )
             => ( Y != produc2761476792215241774st_nat(aTP_Lamm_xw,aa_nat7603221163087113075at_nat(nat_prod_decode,N2)) ) ) ) ) ).

% list_decode.elims
tff(fact_10081_list__decode_Opsimps_I1_J,axiom,
    ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat))
   => ( nat_list_decode(zero_zero_nat) = nil_nat ) ) ).

% list_decode.psimps(1)
tff(fact_10082_list__decode_Osimps_I1_J,axiom,
    nat_list_decode(zero_zero_nat) = nil_nat ).

% list_decode.simps(1)
tff(fact_10083_list__decode_Opelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( nat_list_decode(X) = Y )
     => ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = nil_nat )
             => ~ pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat)) ) )
         => ~ ! [N2: nat] :
                ( ( X = aa_nat_nat(suc,N2) )
               => ( ( Y = produc2761476792215241774st_nat(aTP_Lamm_xw,aa_nat7603221163087113075at_nat(nat_prod_decode,N2)) )
                 => ~ pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N2))) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_10084_unit__factor__simps_I1_J,axiom,
    aa_nat_nat(unit_f2748546683901255202or_nat,zero_zero_nat) = zero_zero_nat ).

% unit_factor_simps(1)
tff(fact_10085_unit__factor__nat__def,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(unit_f2748546683901255202or_nat,N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(unit_f2748546683901255202or_nat,N) = one_one_nat ) ) ) ).

% unit_factor_nat_def
tff(fact_10086_Lcm__eq__0__I__nat,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3))
     => ( aa_set_nat_nat(gcd_Lcm_nat,A3) = zero_zero_nat ) ) ).

% Lcm_eq_0_I_nat
tff(fact_10087_Lcm__0__iff__nat,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( aa_set_nat_nat(gcd_Lcm_nat,A3) = zero_zero_nat )
      <=> pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) ) ) ).

% Lcm_0_iff_nat
tff(fact_10088_Lcm__nat__infinite,axiom,
    ! [M5: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,M5))
     => ( aa_set_nat_nat(gcd_Lcm_nat,M5) = zero_zero_nat ) ) ).

% Lcm_nat_infinite
tff(fact_10089_Lcm__int__greater__eq__0,axiom,
    ! [K4: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),gcd_Lcm_int(K4))) ).

% Lcm_int_greater_eq_0
tff(fact_10090_Lcm__nat__empty,axiom,
    aa_set_nat_nat(gcd_Lcm_nat,bot_bot_set_nat) = one_one_nat ).

% Lcm_nat_empty
tff(fact_10091_Lcm__eq__Max__nat,axiom,
    ! [M5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M5))
     => ( ( M5 != bot_bot_set_nat )
       => ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M5))
         => ( ! [M4: nat,N2: nat] :
                ( pp(aa_set_nat_bool(member_nat(M4),M5))
               => ( pp(aa_set_nat_bool(member_nat(N2),M5))
                 => pp(aa_set_nat_bool(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M4),N2)),M5)) ) )
           => ( aa_set_nat_nat(gcd_Lcm_nat,M5) = aa_set_nat_nat(lattic8265883725875713057ax_nat,M5) ) ) ) ) ) ).

% Lcm_eq_Max_nat
tff(fact_10092_lcm__0__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% lcm_0_iff_nat
tff(fact_10093_lcm__1__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% lcm_1_iff_nat
tff(fact_10094_Lcm__in__lcm__closed__set__nat,axiom,
    ! [M5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M5))
     => ( ( M5 != bot_bot_set_nat )
       => ( ! [M4: nat,N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(M4),M5))
             => ( pp(aa_set_nat_bool(member_nat(N2),M5))
               => pp(aa_set_nat_bool(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M4),N2)),M5)) ) )
         => pp(aa_set_nat_bool(member_nat(aa_set_nat_nat(gcd_Lcm_nat,M5)),M5)) ) ) ) ).

% Lcm_in_lcm_closed_set_nat
tff(fact_10095_lcm__pos__int,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( ( N != zero_zero_int )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N))) ) ) ).

% lcm_pos_int
tff(fact_10096_lcm__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N))) ) ) ).

% lcm_pos_nat
tff(fact_10097_lcm__ge__0__int,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),Y))) ).

% lcm_ge_0_int
tff(fact_10098_lcm__unique__int,axiom,
    ! [D: int,A2: int,B3: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),D))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),D))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B3),D))
        & ! [E3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),E3))
              & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B3),E3)) )
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),E3)) ) )
    <=> ( D = aa_int_int(aa_int_fun_int_int(gcd_lcm_int,A2),B3) ) ) ).

% lcm_unique_int
tff(fact_10099_lcm__cases__int,axiom,
    ! [X: int,Y: int,P: fun_int_bool] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
         => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),Y))) ) )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),aa_int_int(uminus_uminus_int,Y)))) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_int_int(uminus_uminus_int,X)),Y))) ) )
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_int_int(uminus_uminus_int,X)),aa_int_int(uminus_uminus_int,Y)))) ) )
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),Y))) ) ) ) ) ).

% lcm_cases_int
tff(fact_10100_Lcm__nat__def,axiom,
    ! [M5: set_nat] :
      ( ( pp(aa_set_nat_bool(finite_finite_nat,M5))
       => ( aa_set_nat_nat(gcd_Lcm_nat,M5) = aa_set_nat_nat(lattic7826324295020591184_F_nat(gcd_lcm_nat,one_one_nat),M5) ) )
      & ( ~ pp(aa_set_nat_bool(finite_finite_nat,M5))
       => ( aa_set_nat_nat(gcd_Lcm_nat,M5) = zero_zero_nat ) ) ) ).

% Lcm_nat_def
tff(fact_10101_VEBT__internal_Olesseq_Osimps,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,X),Y))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_eq_nat),X),Y)) ) ).

% VEBT_internal.lesseq.simps
tff(fact_10102_VEBT__internal_Olesseq_Oelims_I1_J,axiom,
    ! [X: option_nat,Xa2: option_nat,Y: bool] :
      ( ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,X),Xa2))
      <=> pp(Y) )
     => ( pp(Y)
      <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_eq_nat),X),Xa2)) ) ) ).

% VEBT_internal.lesseq.elims(1)
tff(fact_10103_VEBT__internal_Olesseq_Oelims_I2_J,axiom,
    ! [X: option_nat,Xa2: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,X),Xa2))
     => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_eq_nat),X),Xa2)) ) ).

% VEBT_internal.lesseq.elims(2)
tff(fact_10104_VEBT__internal_Olesseq_Oelims_I3_J,axiom,
    ! [X: option_nat,Xa2: option_nat] :
      ( ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,X),Xa2))
     => ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_eq_nat),X),Xa2)) ) ).

% VEBT_internal.lesseq.elims(3)
tff(fact_10105_VEBT__internal_Oless_Oelims_I3_J,axiom,
    ! [X: option_nat,Xa2: option_nat] :
      ( ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,X),Xa2))
     => ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_nat),X),Xa2)) ) ).

% VEBT_internal.less.elims(3)
tff(fact_10106_VEBT__internal_Oless_Oelims_I2_J,axiom,
    ! [X: option_nat,Xa2: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,X),Xa2))
     => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_nat),X),Xa2)) ) ).

% VEBT_internal.less.elims(2)
tff(fact_10107_VEBT__internal_Oless_Oelims_I1_J,axiom,
    ! [X: option_nat,Xa2: option_nat,Y: bool] :
      ( ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,X),Xa2))
      <=> pp(Y) )
     => ( pp(Y)
      <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_nat),X),Xa2)) ) ) ).

% VEBT_internal.less.elims(1)
tff(fact_10108_VEBT__internal_Oless_Osimps,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,X),Y))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_nat),X),Y)) ) ).

% VEBT_internal.less.simps
tff(fact_10109_VEBT__internal_Ogreater_Osimps,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,X),Y))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(aTP_Lamm_bk),X),Y)) ) ).

% VEBT_internal.greater.simps
tff(fact_10110_VEBT__internal_Ogreater_Oelims_I1_J,axiom,
    ! [X: option_nat,Xa2: option_nat,Y: bool] :
      ( ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,X),Xa2))
      <=> pp(Y) )
     => ( pp(Y)
      <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(aTP_Lamm_bk),X),Xa2)) ) ) ).

% VEBT_internal.greater.elims(1)
tff(fact_10111_VEBT__internal_Ogreater_Oelims_I2_J,axiom,
    ! [X: option_nat,Xa2: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,X),Xa2))
     => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(aTP_Lamm_bk),X),Xa2)) ) ).

% VEBT_internal.greater.elims(2)
tff(fact_10112_VEBT__internal_Ogreater_Oelims_I3_J,axiom,
    ! [X: option_nat,Xa2: option_nat] :
      ( ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,X),Xa2))
     => ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(aTP_Lamm_bk),X),Xa2)) ) ).

% VEBT_internal.greater.elims(3)
tff(fact_10113_VEBT__internal_Ogreater_Opelims_I1_J,axiom,
    ! [X: option_nat,Xa2: option_nat,Y: bool] :
      ( ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,X),Xa2))
      <=> pp(Y) )
     => ( pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_V5711637165310795180er_rel),produc5098337634421038937on_nat(X,Xa2)))
       => ~ ( ( pp(Y)
            <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(aTP_Lamm_bk),X),Xa2)) )
           => ~ pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_V5711637165310795180er_rel),produc5098337634421038937on_nat(X,Xa2))) ) ) ) ).

% VEBT_internal.greater.pelims(1)
tff(fact_10114_VEBT__internal_Ogreater_Ocases,axiom,
    ! [X: produc4953844613479565601on_nat] :
      ~ ! [X5: option_nat,Y3: option_nat] : X != produc5098337634421038937on_nat(X5,Y3) ).

% VEBT_internal.greater.cases
tff(fact_10115_VEBT__internal_Ogreater_Opelims_I3_J,axiom,
    ! [X: option_nat,Xa2: option_nat] :
      ( ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,X),Xa2))
     => ( pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_V5711637165310795180er_rel),produc5098337634421038937on_nat(X,Xa2)))
       => ~ ( pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_V5711637165310795180er_rel),produc5098337634421038937on_nat(X,Xa2)))
           => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(aTP_Lamm_bk),X),Xa2)) ) ) ) ).

% VEBT_internal.greater.pelims(3)
tff(fact_10116_VEBT__internal_Ogreater_Opelims_I2_J,axiom,
    ! [X: option_nat,Xa2: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,X),Xa2))
     => ( pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_V5711637165310795180er_rel),produc5098337634421038937on_nat(X,Xa2)))
       => ~ ( pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_V5711637165310795180er_rel),produc5098337634421038937on_nat(X,Xa2)))
           => ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(aTP_Lamm_bk),X),Xa2)) ) ) ) ).

% VEBT_internal.greater.pelims(2)
tff(fact_10117_VEBT__internal_Olesseq_Opelims_I1_J,axiom,
    ! [X: option_nat,Xa2: option_nat,Y: bool] :
      ( ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,X),Xa2))
      <=> pp(Y) )
     => ( pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_VEBT_lesseq_rel),produc5098337634421038937on_nat(X,Xa2)))
       => ~ ( ( pp(Y)
            <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_eq_nat),X),Xa2)) )
           => ~ pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_VEBT_lesseq_rel),produc5098337634421038937on_nat(X,Xa2))) ) ) ) ).

% VEBT_internal.lesseq.pelims(1)
tff(fact_10118_VEBT__internal_Olesseq_Opelims_I2_J,axiom,
    ! [X: option_nat,Xa2: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,X),Xa2))
     => ( pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_VEBT_lesseq_rel),produc5098337634421038937on_nat(X,Xa2)))
       => ~ ( pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_VEBT_lesseq_rel),produc5098337634421038937on_nat(X,Xa2)))
           => ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_eq_nat),X),Xa2)) ) ) ) ).

% VEBT_internal.lesseq.pelims(2)
tff(fact_10119_VEBT__internal_Olesseq_Opelims_I3_J,axiom,
    ! [X: option_nat,Xa2: option_nat] :
      ( ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,X),Xa2))
     => ( pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_VEBT_lesseq_rel),produc5098337634421038937on_nat(X,Xa2)))
       => ~ ( pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_VEBT_lesseq_rel),produc5098337634421038937on_nat(X,Xa2)))
           => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_eq_nat),X),Xa2)) ) ) ) ).

% VEBT_internal.lesseq.pelims(3)
tff(fact_10120_VEBT__internal_Oless_Opelims_I3_J,axiom,
    ! [X: option_nat,Xa2: option_nat] :
      ( ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,X),Xa2))
     => ( pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_VEBT_less_rel),produc5098337634421038937on_nat(X,Xa2)))
       => ~ ( pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_VEBT_less_rel),produc5098337634421038937on_nat(X,Xa2)))
           => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_nat),X),Xa2)) ) ) ) ).

% VEBT_internal.less.pelims(3)
tff(fact_10121_VEBT__internal_Oless_Opelims_I2_J,axiom,
    ! [X: option_nat,Xa2: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,X),Xa2))
     => ( pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_VEBT_less_rel),produc5098337634421038937on_nat(X,Xa2)))
       => ~ ( pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_VEBT_less_rel),produc5098337634421038937on_nat(X,Xa2)))
           => ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_nat),X),Xa2)) ) ) ) ).

% VEBT_internal.less.pelims(2)
tff(fact_10122_VEBT__internal_Oless_Opelims_I1_J,axiom,
    ! [X: option_nat,Xa2: option_nat,Y: bool] :
      ( ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,X),Xa2))
      <=> pp(Y) )
     => ( pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_VEBT_less_rel),produc5098337634421038937on_nat(X,Xa2)))
       => ~ ( ( pp(Y)
            <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_nat),X),Xa2)) )
           => ~ pp(aa_Pro7647172495381830770t_bool(accp_P8646395344606611882on_nat(vEBT_VEBT_less_rel),produc5098337634421038937on_nat(X,Xa2))) ) ) ) ).

% VEBT_internal.less.pelims(1)
tff(fact_10123_ATP_Olambda__1,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_ub,Uu))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Uu))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).

% ATP.lambda_1
tff(fact_10124_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_wu,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_2
tff(fact_10125_ATP_Olambda__3,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_tl,Uu) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),one_one_real) ).

% ATP.lambda_3
tff(fact_10126_ATP_Olambda__4,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_ti,Uu) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),one_one_rat) ).

% ATP.lambda_4
tff(fact_10127_ATP_Olambda__5,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_tj,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_5
tff(fact_10128_ATP_Olambda__6,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_tk,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_6
tff(fact_10129_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_xg,Uu) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),zero_zero_nat) ).

% ATP.lambda_7
tff(fact_10130_ATP_Olambda__8,axiom,
    ! [Uu: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_xf,Uu))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu)),product_snd_int_int(Uu)))) ) ).

% ATP.lambda_8
tff(fact_10131_ATP_Olambda__9,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ur,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_gv,Uu)) ).

% ATP.lambda_9
tff(fact_10132_ATP_Olambda__10,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_vj,Uu) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_tn,Uu)) ).

% ATP.lambda_10
tff(fact_10133_ATP_Olambda__11,axiom,
    ! [Uu: real] : aa_rea5117059003799598681omplex(aTP_Lamm_wd,Uu) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc9214369023355538061x_bool(aTP_Lamm_wc(Uu)))) ).

% ATP.lambda_11
tff(fact_10134_ATP_Olambda__12,axiom,
    ! [Uu: real] : aa_rea5428163979925208277l_real(aTP_Lamm_wb,Uu) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5257810830037637649l_bool(aTP_Lamm_wa(Uu)))) ).

% ATP.lambda_12
tff(fact_10135_ATP_Olambda__13,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_tu,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_13
tff(fact_10136_ATP_Olambda__14,axiom,
    ! [Uu: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aTP_Lamm_xe,Uu))
    <=> ? [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
          & ? [K3: nat] :
            ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(Uu,N3))) ) ) ) ).

% ATP.lambda_14
tff(fact_10137_ATP_Olambda__15,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_wj,Uu))
    <=> ? [I4: int,N3: nat] :
          ( ( Uu = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I4)),aa_nat_real(semiri5074537144036343181t_real,N3)) )
          & ( N3 != zero_zero_nat ) ) ) ).

% ATP.lambda_15
tff(fact_10138_ATP_Olambda__16,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_uj,Uu),Uua) = aa_int_int(aa_int_fun_int_int(if_int(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu))),one_one_int)) ).

% ATP.lambda_16
tff(fact_10139_ATP_Olambda__17,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_um,Uu),Uua) = aa_num_num(aa_num_fun_num_num(if_num(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger)),aa_num_num(aa_num_fun_num_num(plus_plus_num,code_num_of_integer(Uu)),code_num_of_integer(Uu))),aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,code_num_of_integer(Uu)),code_num_of_integer(Uu))),one)) ).

% ATP.lambda_17
tff(fact_10140_ATP_Olambda__18,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_un,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,code_nat_of_integer(Uu)),code_nat_of_integer(Uu))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,code_nat_of_integer(Uu)),code_nat_of_integer(Uu))),one_one_nat)) ).

% ATP.lambda_18
tff(fact_10141_ATP_Olambda__19,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_xq(Uu),Uua) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_xp(Uua),aa_Ext7969882492699152700d_enat(if_Extended_enat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),zero_z5237406670263579293d_enat),extend5688581933313929465d_enat)),Uu) ).

% ATP.lambda_19
tff(fact_10142_ATP_Olambda__20,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( pp(aa_nat_bool(aa_Ext6309394871002001526t_bool(aTP_Lamm_xl,Uu),Uua))
    <=> pp(aa_Ext2772979874139974381t_bool(aa_boo1364755444788961487t_bool(extend8126323780198025762t_bool(aa_nat_fun_nat_bool(aTP_Lamm_bl,Uua)),fFalse),Uu)) ) ).

% ATP.lambda_20
tff(fact_10143_ATP_Olambda__21,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( pp(aa_nat_bool(aa_Ext6309394871002001526t_bool(aTP_Lamm_xm,Uu),Uua))
    <=> pp(aa_Ext2772979874139974381t_bool(aa_boo1364755444788961487t_bool(extend8126323780198025762t_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua)),fTrue),Uu)) ) ).

% ATP.lambda_21
tff(fact_10144_ATP_Olambda__22,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_gu,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_22
tff(fact_10145_ATP_Olambda__23,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_uc,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
        & ( aa_real_real(sin_real,Uua) = Uu ) ) ) ).

% ATP.lambda_23
tff(fact_10146_ATP_Olambda__24,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_tz,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
        & ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).

% ATP.lambda_24
tff(fact_10147_ATP_Olambda__25,axiom,
    ! [Uu: complex,Uua: real] :
      ( pp(aa_real_bool(aa_com8943241093633526360l_bool(aTP_Lamm_wl,Uu),Uua))
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu) = aa_real_complex(cis,Uua) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),pi)) ) ) ).

% ATP.lambda_25
tff(fact_10148_ATP_Olambda__26,axiom,
    ! [Uu: real,Uua: int] :
      ( pp(aa_int_bool(aa_real_fun_int_bool(aTP_Lamm_ud,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uu),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int)))) ) ) ).

% ATP.lambda_26
tff(fact_10149_ATP_Olambda__27,axiom,
    ! [Uu: rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ue(Uu),Uua))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uu),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int)))) ) ) ).

% ATP.lambda_27
tff(fact_10150_ATP_Olambda__28,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_gv,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_28
tff(fact_10151_ATP_Olambda__29,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_us,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_29
tff(fact_10152_ATP_Olambda__30,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vl(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_30
tff(fact_10153_ATP_Olambda__31,axiom,
    ! [Uu: rat,Uua: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_xd(Uu),Uua))
    <=> ( ( Uu = aa_int_rat(fract(product_fst_int_int(Uua)),product_snd_int_int(Uua)) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(Uua)))
        & pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(Uua)),product_snd_int_int(Uua))) ) ) ).

% ATP.lambda_31
tff(fact_10154_ATP_Olambda__32,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_qo(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu),Uua)),aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Uu),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_32
tff(fact_10155_ATP_Olambda__33,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qp,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu),Uua)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_33
tff(fact_10156_ATP_Olambda__34,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qn(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Uu),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_34
tff(fact_10157_ATP_Olambda__35,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_wv,Uu),Uua))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uu),Uua))
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_35
tff(fact_10158_ATP_Olambda__36,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_vz(Uu),Uua))
    <=> ( pp(aa_set_int_bool(member_int(Uua),Uu))
        & ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),Uu))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Uua)) ) ) ) ).

% ATP.lambda_36
tff(fact_10159_ATP_Olambda__37,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_th,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_37
tff(fact_10160_ATP_Olambda__38,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wh,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uua)),Uua) ).

% ATP.lambda_38
tff(fact_10161_ATP_Olambda__39,axiom,
    ! [Uu: nat,Uua: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_bs,Uu),Uua))
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_39
tff(fact_10162_ATP_Olambda__40,axiom,
    ! [Uu: nat,Uua: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aTP_Lamm_gc,Uu),Uua))
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_40
tff(fact_10163_ATP_Olambda__41,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_fy,Uu),Uua))
    <=> ( pp(aa_set_real_bool(member_real(Uua),ring_1_Ints_real))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Uua)),Uu)) ) ) ).

% ATP.lambda_41
tff(fact_10164_ATP_Olambda__42,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_fz,Uu),Uua))
    <=> ( pp(aa_set_rat_bool(member_rat(Uua),ring_1_Ints_rat))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Uua)),Uu)) ) ) ).

% ATP.lambda_42
tff(fact_10165_ATP_Olambda__43,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_gt,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_43
tff(fact_10166_ATP_Olambda__44,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ua,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),pi))
        & ( aa_real_real(cos_real,Uua) = Uu ) ) ) ).

% ATP.lambda_44
tff(fact_10167_ATP_Olambda__45,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_qa(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_45
tff(fact_10168_ATP_Olambda__46,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_px(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_46
tff(fact_10169_ATP_Olambda__47,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_pz(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_47
tff(fact_10170_ATP_Olambda__48,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_py(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_48
tff(fact_10171_ATP_Olambda__49,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pg(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_49
tff(fact_10172_ATP_Olambda__50,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_pe(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_50
tff(fact_10173_ATP_Olambda__51,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_pf(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,aa_nat_nat(suc,Uua))),aa_nat_int(Uu,Uua)) ).

% ATP.lambda_51
tff(fact_10174_ATP_Olambda__52,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_gw(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_52
tff(fact_10175_ATP_Olambda__53,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_gx(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_53
tff(fact_10176_ATP_Olambda__54,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kn(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_54
tff(fact_10177_ATP_Olambda__55,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hs(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)),aa_nat_int(power_power_int(zero_zero_int),Uua)) ).

% ATP.lambda_55
tff(fact_10178_ATP_Olambda__56,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pw(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_56
tff(fact_10179_ATP_Olambda__57,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_pu(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_57
tff(fact_10180_ATP_Olambda__58,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_pv(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_58
tff(fact_10181_ATP_Olambda__59,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pt(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_59
tff(fact_10182_ATP_Olambda__60,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_pr(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_60
tff(fact_10183_ATP_Olambda__61,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ps(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_61
tff(fact_10184_ATP_Olambda__62,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_so(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_62
tff(fact_10185_ATP_Olambda__63,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_sm(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_63
tff(fact_10186_ATP_Olambda__64,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_sn(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_64
tff(fact_10187_ATP_Olambda__65,axiom,
    ! [Uu: fun_Ex1451620480498224165t_real,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aTP_Lamm_np(Uu),Uua))
    <=> ( aa_Ext2541572767299617110t_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_65
tff(fact_10188_ATP_Olambda__66,axiom,
    ! [Uu: fun_Ex4152069839811072081at_rat,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aTP_Lamm_nt(Uu),Uua))
    <=> ( aa_Extended_enat_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_66
tff(fact_10189_ATP_Olambda__67,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_no(Uu),Uua))
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_67
tff(fact_10190_ATP_Olambda__68,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_ns(Uu),Uua))
    <=> ( aa_complex_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_68
tff(fact_10191_ATP_Olambda__69,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_nm(Uu),Uua))
    <=> ( aa_real_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_69
tff(fact_10192_ATP_Olambda__70,axiom,
    ! [Uu: fun_real_rat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_nq(Uu),Uua))
    <=> ( aa_real_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_70
tff(fact_10193_ATP_Olambda__71,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_nu(Uu),Uua))
    <=> ( aa_real_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_71
tff(fact_10194_ATP_Olambda__72,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_nn(Uu),Uua))
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_72
tff(fact_10195_ATP_Olambda__73,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_nr(Uu),Uua))
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_73
tff(fact_10196_ATP_Olambda__74,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_nv(Uu),Uua))
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_74
tff(fact_10197_ATP_Olambda__75,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vt(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vl(Uu)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_75
tff(fact_10198_ATP_Olambda__76,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vs(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vl(Uu)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_76
tff(fact_10199_ATP_Olambda__77,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_jr(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Uua))),aa_nat_complex(power_power_complex(Uu),Uua)) ).

% ATP.lambda_77
tff(fact_10200_ATP_Olambda__78,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jq,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_78
tff(fact_10201_ATP_Olambda__79,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_uq,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_79
tff(fact_10202_ATP_Olambda__80,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tv,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_80
tff(fact_10203_ATP_Olambda__81,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tw,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_81
tff(fact_10204_ATP_Olambda__82,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_qc(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu) ).

% ATP.lambda_82
tff(fact_10205_ATP_Olambda__83,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aTP_Lamm_qe(Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu) ).

% ATP.lambda_83
tff(fact_10206_ATP_Olambda__84,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_qd(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).

% ATP.lambda_84
tff(fact_10207_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_up(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(Uu),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_85
tff(fact_10208_ATP_Olambda__86,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_list_nat(aa_nat6157064778417537467st_nat(aTP_Lamm_xw,Uu),Uua) = cons_nat(Uu,nat_list_decode(Uua)) ).

% ATP.lambda_86
tff(fact_10209_ATP_Olambda__87,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aTP_Lamm_ba(Uu),Uua))
    <=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(Uua),Uu)) ) ).

% ATP.lambda_87
tff(fact_10210_ATP_Olambda__88,axiom,
    ! [Uu: set_Extended_enat,Uua: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(aTP_Lamm_bb(Uu),Uua))
    <=> pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(Uua),Uu)) ) ).

% ATP.lambda_88
tff(fact_10211_ATP_Olambda__89,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( pp(aa_set_complex_bool(aTP_Lamm_az(Uu),Uua))
    <=> pp(aa_set_complex_bool(ord_le211207098394363844omplex(Uua),Uu)) ) ).

% ATP.lambda_89
tff(fact_10212_ATP_Olambda__90,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_ay,Uu),Uua))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Uua),Uu)) ) ).

% ATP.lambda_90
tff(fact_10213_ATP_Olambda__91,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( pp(aa_set_int_bool(aTP_Lamm_bc(Uu),Uua))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(Uua),Uu)) ) ).

% ATP.lambda_91
tff(fact_10214_ATP_Olambda__92,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bl,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uu)) ) ).

% ATP.lambda_92
tff(fact_10215_ATP_Olambda__93,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uo,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_93
tff(fact_10216_ATP_Olambda__94,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_sj,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),Uu)) ) ).

% ATP.lambda_94
tff(fact_10217_ATP_Olambda__95,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_sf,Uu),Uua))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,Uua),Uu)) ) ).

% ATP.lambda_95
tff(fact_10218_ATP_Olambda__96,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_sg,Uu),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uua),Uu)) ) ).

% ATP.lambda_96
tff(fact_10219_ATP_Olambda__97,axiom,
    ! [Uu: num,Uua: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_sh,Uu),Uua))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Uua),Uu)) ) ).

% ATP.lambda_97
tff(fact_10220_ATP_Olambda__98,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bk,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu)) ) ).

% ATP.lambda_98
tff(fact_10221_ATP_Olambda__99,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_si,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uu)) ) ).

% ATP.lambda_99
tff(fact_10222_ATP_Olambda__100,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vm,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_100
tff(fact_10223_ATP_Olambda__101,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ul,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_101
tff(fact_10224_ATP_Olambda__102,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ut,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_102
tff(fact_10225_ATP_Olambda__103,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wt,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_103
tff(fact_10226_ATP_Olambda__104,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_uk,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_104
tff(fact_10227_ATP_Olambda__105,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_uu,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu) ).

% ATP.lambda_105
tff(fact_10228_ATP_Olambda__106,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_tn,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),Uu)) ) ).

% ATP.lambda_106
tff(fact_10229_ATP_Olambda__107,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vp,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_107
tff(fact_10230_ATP_Olambda__108,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: produc859450856879609959at_nat] :
      ( pp(aa_Pro6098352494050694700t_bool(aTP_Lamm_cm(Uu),Uua))
    <=> member8206827879206165904at_nat(Uua,Uu) ) ).

% ATP.lambda_108
tff(fact_10231_ATP_Olambda__109,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: produc3843707927480180839at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(aTP_Lamm_cn(Uu),Uua))
    <=> member8757157785044589968at_nat(Uua,Uu) ) ).

% ATP.lambda_109
tff(fact_10232_ATP_Olambda__110,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aTP_Lamm_fv(Uu),Uua))
    <=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Uua),Uu)) ) ).

% ATP.lambda_110
tff(fact_10233_ATP_Olambda__111,axiom,
    ! [Uu: set_Extended_enat,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aTP_Lamm_xo(Uu),Uua))
    <=> pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uua),Uu)) ) ).

% ATP.lambda_111
tff(fact_10234_ATP_Olambda__112,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_aa(Uu),Uua))
    <=> pp(aa_set_real_bool(member_real(Uua),Uu)) ) ).

% ATP.lambda_112
tff(fact_10235_ATP_Olambda__113,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_ab(Uu),Uua))
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_113
tff(fact_10236_ATP_Olambda__114,axiom,
    ! [Uu: set_bool,Uua: bool] :
      ( pp(aa_bool_bool(aTP_Lamm_a(Uu),Uua))
    <=> pp(aa_set_bool_bool(member_bool(Uua),Uu)) ) ).

% ATP.lambda_114
tff(fact_10237_ATP_Olambda__115,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_ac(Uu),Uua))
    <=> pp(aa_set_set_nat_bool(member_set_nat(Uua),Uu)) ) ).

% ATP.lambda_115
tff(fact_10238_ATP_Olambda__116,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_ad(Uu),Uua))
    <=> pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).

% ATP.lambda_116
tff(fact_10239_ATP_Olambda__117,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ae(Uu),Uua))
    <=> pp(aa_set_int_bool(member_int(Uua),Uu)) ) ).

% ATP.lambda_117
tff(fact_10240_ATP_Olambda__118,axiom,
    ! [Uu: produc859450856879609959at_nat,Uua: produc859450856879609959at_nat] :
      ( pp(aa_Pro6098352494050694700t_bool(aTP_Lamm_cq(Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_118
tff(fact_10241_ATP_Olambda__119,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: produc3843707927480180839at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(aTP_Lamm_cr(Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_119
tff(fact_10242_ATP_Olambda__120,axiom,
    ! [Uu: product_prod_nat_nat,Uua: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(aTP_Lamm_bd,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_120
tff(fact_10243_ATP_Olambda__121,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_bg,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_121
tff(fact_10244_ATP_Olambda__122,axiom,
    ! [Uu: list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aTP_Lamm_be,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_122
tff(fact_10245_ATP_Olambda__123,axiom,
    ! [Uu: bool,Uua: bool] :
      ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_bh,Uu),Uua))
    <=> ( pp(Uua)
      <=> pp(Uu) ) ) ).

% ATP.lambda_123
tff(fact_10246_ATP_Olambda__124,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_bf,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_124
tff(fact_10247_ATP_Olambda__125,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bi,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_125
tff(fact_10248_ATP_Olambda__126,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_bj,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_126
tff(fact_10249_ATP_Olambda__127,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_xk(Uu),Uua) = order_underS_nat(bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_127
tff(fact_10250_ATP_Olambda__128,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hv(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_128
tff(fact_10251_ATP_Olambda__129,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_uf(Uu),Uua))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_129
tff(fact_10252_ATP_Olambda__130,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_pd(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_130
tff(fact_10253_ATP_Olambda__131,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_nw(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_131
tff(fact_10254_ATP_Olambda__132,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ny(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_132
tff(fact_10255_ATP_Olambda__133,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_wp,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_wo(Uu),Uua)) ).

% ATP.lambda_133
tff(fact_10256_ATP_Olambda__134,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_wn,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_wm(Uu),Uua)) ).

% ATP.lambda_134
tff(fact_10257_ATP_Olambda__135,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_vg(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_vf(Uu),Uua)) ).

% ATP.lambda_135
tff(fact_10258_ATP_Olambda__136,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_wk(Uu),Uua) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_136
tff(fact_10259_ATP_Olambda__137,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_xi(Uu),Uua) = aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uua)) ).

% ATP.lambda_137
tff(fact_10260_ATP_Olambda__138,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vr,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_138
tff(fact_10261_ATP_Olambda__139,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_xp(Uu),Uua) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uua)) ).

% ATP.lambda_139
tff(fact_10262_ATP_Olambda__140,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xv,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_140
tff(fact_10263_ATP_Olambda__141,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xu,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_141
tff(fact_10264_ATP_Olambda__142,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xs,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_142
tff(fact_10265_ATP_Olambda__143,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xt,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_143
tff(fact_10266_ATP_Olambda__144,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ct(Uu),Uua))
    <=> ~ pp(aa_set_real_bool(member_real(Uua),Uu)) ) ).

% ATP.lambda_144
tff(fact_10267_ATP_Olambda__145,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_cu(Uu),Uua))
    <=> ~ member_list_nat(Uua,Uu) ) ).

% ATP.lambda_145
tff(fact_10268_ATP_Olambda__146,axiom,
    ! [Uu: set_bool,Uua: bool] :
      ( pp(aa_bool_bool(aTP_Lamm_cs(Uu),Uua))
    <=> ~ pp(aa_set_bool_bool(member_bool(Uua),Uu)) ) ).

% ATP.lambda_146
tff(fact_10269_ATP_Olambda__147,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_cv(Uu),Uua))
    <=> ~ pp(aa_set_set_nat_bool(member_set_nat(Uua),Uu)) ) ).

% ATP.lambda_147
tff(fact_10270_ATP_Olambda__148,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_cw(Uu),Uua))
    <=> ~ pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).

% ATP.lambda_148
tff(fact_10271_ATP_Olambda__149,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_cx(Uu),Uua))
    <=> ~ pp(aa_set_int_bool(member_int(Uua),Uu)) ) ).

% ATP.lambda_149
tff(fact_10272_ATP_Olambda__150,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_real(aTP_Lamm_ml(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_150
tff(fact_10273_ATP_Olambda__151,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_ht(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_151
tff(fact_10274_ATP_Olambda__152,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hw(Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_152
tff(fact_10275_ATP_Olambda__153,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vo(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_153
tff(fact_10276_ATP_Olambda__154,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_ta(Uu),Uua) = aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_154
tff(fact_10277_ATP_Olambda__155,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_sx(Uu),Uua) = aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_155
tff(fact_10278_ATP_Olambda__156,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_sz(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_156
tff(fact_10279_ATP_Olambda__157,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_sy(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_157
tff(fact_10280_ATP_Olambda__158,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_vv(Uu),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_158
tff(fact_10281_ATP_Olambda__159,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hu(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_159
tff(fact_10282_ATP_Olambda__160,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_km(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_160
tff(fact_10283_ATP_Olambda__161,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_vi(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_161
tff(fact_10284_ATP_Olambda__162,axiom,
    ! [Uu: fun_real_bool,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_cy(Uu),Uua))
    <=> ~ pp(aa_real_bool(Uu,Uua)) ) ).

% ATP.lambda_162
tff(fact_10285_ATP_Olambda__163,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_cz(Uu),Uua))
    <=> ~ pp(aa_list_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_163
tff(fact_10286_ATP_Olambda__164,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_da(Uu),Uua))
    <=> ~ pp(aa_set_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_164
tff(fact_10287_ATP_Olambda__165,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_db(Uu),Uua))
    <=> ~ pp(aa_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_165
tff(fact_10288_ATP_Olambda__166,axiom,
    ! [Uu: fun_int_bool,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_dc(Uu),Uua))
    <=> ~ pp(aa_int_bool(Uu,Uua)) ) ).

% ATP.lambda_166
tff(fact_10289_ATP_Olambda__167,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_xj(Uu),Uua) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bk,Uu)) ).

% ATP.lambda_167
tff(fact_10290_ATP_Olambda__168,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_vy(Uu),Uua))
    <=> ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),Uu))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Uua)) ) ) ).

% ATP.lambda_168
tff(fact_10291_ATP_Olambda__169,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_uh(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,numera6620942414471956472nteger(Uu))),produc1086072967326762835nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Uua),Uub)) ).

% ATP.lambda_169
tff(fact_10292_ATP_Olambda__170,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ts(Uu),Uua),Uub) = aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),Uub)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu)))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_170
tff(fact_10293_ATP_Olambda__171,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_tt(Uu),Uua),Uub) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu))),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_171
tff(fact_10294_ATP_Olambda__172,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_hh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_set_nat_bool(member_nat(Uub),Uu)),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_172
tff(fact_10295_ATP_Olambda__173,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hi(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_set_nat_bool(member_nat(Uub),Uu)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_173
tff(fact_10296_ATP_Olambda__174,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_hj(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(if_int(aa_set_nat_bool(member_nat(Uub),Uu)),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_174
tff(fact_10297_ATP_Olambda__175,axiom,
    ! [Uu: fun_Ex1451620480498224165t_real,Uua: set_Extended_enat,Uub: extended_enat] : aa_Ext2541572767299617110t_real(aa_set3355554880467357083t_real(aTP_Lamm_ng(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uua)),aa_Ext2541572767299617110t_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_175
tff(fact_10298_ATP_Olambda__176,axiom,
    ! [Uu: fun_Ex4152069839811072081at_rat,Uua: set_Extended_enat,Uub: extended_enat] : aa_Extended_enat_rat(aa_set8847123550896565447at_rat(aTP_Lamm_nl(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uua)),aa_Extended_enat_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_176
tff(fact_10299_ATP_Olambda__177,axiom,
    ! [Uu: fun_complex_real,Uua: set_complex,Uub: complex] : aa_complex_real(aa_set904117813986905849x_real(aTP_Lamm_nf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_set_complex_bool(member_complex(Uub),Uua)),aa_complex_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_177
tff(fact_10300_ATP_Olambda__178,axiom,
    ! [Uu: fun_complex_rat,Uua: set_complex,Uub: complex] : aa_complex_rat(aa_set7482584248617139493ex_rat(aTP_Lamm_nk(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_set_complex_bool(member_complex(Uub),Uua)),aa_complex_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_178
tff(fact_10301_ATP_Olambda__179,axiom,
    ! [Uu: fun_real_real,Uua: set_real,Uub: real] : aa_real_real(aa_set8744029839670122869l_real(aTP_Lamm_nc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_set_real_bool(member_real(Uub),Uua)),aa_real_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_179
tff(fact_10302_ATP_Olambda__180,axiom,
    ! [Uu: fun_real_rat,Uua: set_real,Uub: real] : aa_real_rat(aa_set4110029769364456865al_rat(aTP_Lamm_nh(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_set_real_bool(member_real(Uub),Uua)),aa_real_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_180
tff(fact_10303_ATP_Olambda__181,axiom,
    ! [Uu: fun_bool_real,Uua: set_bool,Uub: bool] : aa_bool_real(aa_set4485503625295242075l_real(aTP_Lamm_nd(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_set_bool_bool(member_bool(Uub),Uua)),aa_bool_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_181
tff(fact_10304_ATP_Olambda__182,axiom,
    ! [Uu: fun_bool_rat,Uua: set_bool,Uub: bool] : aa_bool_rat(aa_set1787953457065026183ol_rat(aTP_Lamm_ni(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_set_bool_bool(member_bool(Uub),Uua)),aa_bool_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_182
tff(fact_10305_ATP_Olambda__183,axiom,
    ! [Uu: fun_int_real,Uua: set_int,Uub: int] : aa_int_real(aa_set6366552415685541621t_real(aTP_Lamm_ne(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_set_int_bool(member_int(Uub),Uua)),aa_int_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_183
tff(fact_10306_ATP_Olambda__184,axiom,
    ! [Uu: fun_int_rat,Uua: set_int,Uub: int] : aa_int_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_nj(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_set_int_bool(member_int(Uub),Uua)),aa_int_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_184
tff(fact_10307_ATP_Olambda__185,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1749635019461158908t_real(aTP_Lamm_jw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uu),Uub)),aa_Ext2541572767299617110t_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_185
tff(fact_10308_ATP_Olambda__186,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_jv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub)),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_186
tff(fact_10309_ATP_Olambda__187,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_kb(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub)),aa_complex_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_187
tff(fact_10310_ATP_Olambda__188,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_js(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub)),aa_real_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_188
tff(fact_10311_ATP_Olambda__189,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_jx(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub)),aa_real_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_189
tff(fact_10312_ATP_Olambda__190,axiom,
    ! [Uu: bool,Uua: fun_bool_real,Uub: bool] : aa_bool_real(aa_fun8933258245593490108l_real(aTP_Lamm_jt(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,Uu),Uub)),aa_bool_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_190
tff(fact_10313_ATP_Olambda__191,axiom,
    ! [Uu: bool,Uua: fun_bool_rat,Uub: bool] : aa_bool_rat(aa_fun224000846028461884ol_rat(aTP_Lamm_jy(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,Uu),Uub)),aa_bool_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_191
tff(fact_10314_ATP_Olambda__192,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_jz(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub)),aa_nat_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_192
tff(fact_10315_ATP_Olambda__193,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ju(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub)),aa_int_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_193
tff(fact_10316_ATP_Olambda__194,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ka(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub)),aa_int_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_194
tff(fact_10317_ATP_Olambda__195,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1749635019461158908t_real(aTP_Lamm_kg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uub),Uu)),aa_Ext2541572767299617110t_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_195
tff(fact_10318_ATP_Olambda__196,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_kf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu)),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_196
tff(fact_10319_ATP_Olambda__197,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_kl(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu)),aa_complex_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_197
tff(fact_10320_ATP_Olambda__198,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_kc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu)),aa_real_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_198
tff(fact_10321_ATP_Olambda__199,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_kh(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu)),aa_real_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_199
tff(fact_10322_ATP_Olambda__200,axiom,
    ! [Uu: bool,Uua: fun_bool_real,Uub: bool] : aa_bool_real(aa_fun8933258245593490108l_real(aTP_Lamm_kd(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,Uub),Uu)),aa_bool_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_200
tff(fact_10323_ATP_Olambda__201,axiom,
    ! [Uu: bool,Uua: fun_bool_rat,Uub: bool] : aa_bool_rat(aa_fun224000846028461884ol_rat(aTP_Lamm_ki(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,Uub),Uu)),aa_bool_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_201
tff(fact_10324_ATP_Olambda__202,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ha(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_202
tff(fact_10325_ATP_Olambda__203,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_kj(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_203
tff(fact_10326_ATP_Olambda__204,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hb(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_204
tff(fact_10327_ATP_Olambda__205,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_hc(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_205
tff(fact_10328_ATP_Olambda__206,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ke(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu)),aa_int_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_206
tff(fact_10329_ATP_Olambda__207,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_kk(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu)),aa_int_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_207
tff(fact_10330_ATP_Olambda__208,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_wg(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu),Uub))) ).

% ATP.lambda_208
tff(fact_10331_ATP_Olambda__209,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_wf(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu,Uub))) ).

% ATP.lambda_209
tff(fact_10332_ATP_Olambda__210,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_hk(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(Uu,Uub)),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_210
tff(fact_10333_ATP_Olambda__211,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hl(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(Uu,Uub)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_211
tff(fact_10334_ATP_Olambda__212,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_hm(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(if_int(aa_nat_bool(Uu,Uub)),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_212
tff(fact_10335_ATP_Olambda__213,axiom,
    ! [Uu: fun_Ex1451620480498224165t_real,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1090353380788292083t_real(aTP_Lamm_ms(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Ext2772979874139974381t_bool(Uua,Uub)),aa_Ext2541572767299617110t_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_213
tff(fact_10336_ATP_Olambda__214,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_bool,Uub: complex] : aa_complex_real(aa_fun2902006202505214429x_real(aTP_Lamm_mq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(Uua,Uub)),aa_complex_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_214
tff(fact_10337_ATP_Olambda__215,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_bool,Uub: complex] : aa_complex_rat(aa_fun5529999419959482889ex_rat(aTP_Lamm_mx(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_complex_bool(Uua,Uub)),aa_complex_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_215
tff(fact_10338_ATP_Olambda__216,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_mn(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(Uua,Uub)),aa_real_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_216
tff(fact_10339_ATP_Olambda__217,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_bool,Uub: real] : aa_real_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_mu(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_real_bool(Uua,Uub)),aa_real_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_217
tff(fact_10340_ATP_Olambda__218,axiom,
    ! [Uu: fun_bool_real,Uua: fun_bool_bool,Uub: bool] : aa_bool_real(aa_fun316222562968675029l_real(aTP_Lamm_mm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_bool_bool(Uua,Uub)),aa_bool_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_218
tff(fact_10341_ATP_Olambda__219,axiom,
    ! [Uu: fun_bool_rat,Uua: fun_bool_bool,Uub: bool] : aa_bool_rat(aa_fun4278745445771542273ol_rat(aTP_Lamm_mt(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_bool_bool(Uua,Uub)),aa_bool_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_219
tff(fact_10342_ATP_Olambda__220,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_bool,Uub: nat] : aa_nat_rat(aa_fun5656162108953708681at_rat(aTP_Lamm_mv(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_nat_bool(Uua,Uub)),aa_nat_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_220
tff(fact_10343_ATP_Olambda__221,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_mo(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(Uua,Uub)),aa_int_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_221
tff(fact_10344_ATP_Olambda__222,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_bool,Uub: int] : aa_int_rat(aa_fun4400602662665189257nt_rat(aTP_Lamm_mw(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_int_bool(Uua,Uub)),aa_int_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_222
tff(fact_10345_ATP_Olambda__223,axiom,
    ! [Uu: fun_Ex6753287174829011282omplex,Uua: complex,Uub: extended_enat] : aa_Ext8748583412541189208omplex(aa_com6527309619268954171omplex(aTP_Lamm_mi(Uu),Uua),Uub) = aa_complex_complex(aa_Ext6914378186415866627omplex(Uu,Uub),Uua) ).

% ATP.lambda_223
tff(fact_10346_ATP_Olambda__224,axiom,
    ! [Uu: fun_Ex7123704652682714774at_nat,Uua: nat,Uub: extended_enat] : aa_Extended_enat_nat(aa_nat887872048497836987at_nat(aTP_Lamm_ll(Uu),Uua),Uub) = aa_nat_nat(aa_Ext158989295719858759at_nat(Uu,Uub),Uua) ).

% ATP.lambda_224
tff(fact_10347_ATP_Olambda__225,axiom,
    ! [Uu: fun_co636942507509095880at_nat,Uua: nat,Uub: complex] : aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_lg(Uu),Uua),Uub) = aa_nat_nat(aa_com6383720667567721865at_nat(Uu,Uub),Uua) ).

% ATP.lambda_225
tff(fact_10348_ATP_Olambda__226,axiom,
    ! [Uu: fun_re7208826350674946562omplex,Uua: complex,Uub: real] : aa_real_complex(aa_com4766338864777664963omplex(aTP_Lamm_lu(Uu),Uua),Uub) = aa_complex_complex(aa_rea2618301233319580611omplex(Uu,Uub),Uua) ).

% ATP.lambda_226
tff(fact_10349_ATP_Olambda__227,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_vh(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_227
tff(fact_10350_ATP_Olambda__228,axiom,
    ! [Uu: fun_real_fun_nat_nat,Uua: nat,Uub: real] : aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_kx(Uu),Uua),Uub) = aa_nat_nat(aa_real_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_228
tff(fact_10351_ATP_Olambda__229,axiom,
    ! [Uu: fun_bo6990880009230973955omplex,Uua: complex,Uub: bool] : aa_bool_complex(aa_com2616170237055825644omplex(aTP_Lamm_lq(Uu),Uua),Uub) = aa_complex_complex(aa_boo4396326699157026228omplex(Uu,Uub),Uua) ).

% ATP.lambda_229
tff(fact_10352_ATP_Olambda__230,axiom,
    ! [Uu: fun_bool_fun_nat_nat,Uua: nat,Uub: bool] : aa_bool_nat(aa_nat_fun_bool_nat(aTP_Lamm_kt(Uu),Uua),Uub) = aa_nat_nat(aa_bool_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_230
tff(fact_10353_ATP_Olambda__231,axiom,
    ! [Uu: fun_na1591161823130732966omplex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ly(Uu),Uua),Uub) = aa_complex_complex(aa_nat3829984526574172007omplex(Uu,Uub),Uua) ).

% ATP.lambda_231
tff(fact_10354_ATP_Olambda__232,axiom,
    ! [Uu: fun_in1307110748760623106omplex,Uua: complex,Uub: int] : aa_int_complex(aa_com9156536935073639875omplex(aTP_Lamm_md(Uu),Uua),Uub) = aa_complex_complex(aa_int3545933452204062147omplex(Uu,Uub),Uua) ).

% ATP.lambda_232
tff(fact_10355_ATP_Olambda__233,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_lb(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_233
tff(fact_10356_ATP_Olambda__234,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_pa(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),one_one_complex),zero_zero_complex)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_234
tff(fact_10357_ATP_Olambda__235,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),one_one_real),zero_zero_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_235
tff(fact_10358_ATP_Olambda__236,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_pc(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),one_one_int),zero_zero_int)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_236
tff(fact_10359_ATP_Olambda__237,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod2825494507867081354r_bool(aa_Cod4981436349852597092r_bool(aTP_Lamm_we(Uu),Uua),Uub) = produc5135875973608410785r_bool(if_Code_integer(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),one_one_Code_integer)) ).

% ATP.lambda_237
tff(fact_10360_ATP_Olambda__238,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_xc(Uu),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_xb(Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_238
tff(fact_10361_ATP_Olambda__239,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_xa(Uu),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_wz(Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_239
tff(fact_10362_ATP_Olambda__240,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ux(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_240
tff(fact_10363_ATP_Olambda__241,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_uy(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_241
tff(fact_10364_ATP_Olambda__242,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_kp(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(zero_zero_complex),Uub))),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_242
tff(fact_10365_ATP_Olambda__243,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_kq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_243
tff(fact_10366_ATP_Olambda__244,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ko(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub))),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_244
tff(fact_10367_ATP_Olambda__245,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ve(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_245
tff(fact_10368_ATP_Olambda__246,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_tm(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_246
tff(fact_10369_ATP_Olambda__247,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_wr(Uu),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_247
tff(fact_10370_ATP_Olambda__248,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_sc(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),aa_nat_complex(Uu,Uub))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_248
tff(fact_10371_ATP_Olambda__249,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_se(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),aa_nat_real(Uu,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_249
tff(fact_10372_ATP_Olambda__250,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_ws(Uu),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_250
tff(fact_10373_ATP_Olambda__251,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_gp(Uu),Uua),Uub))
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua)) ) ) ).

% ATP.lambda_251
tff(fact_10374_ATP_Olambda__252,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( pp(aa_lis6649954313659033868t_bool(aa_nat1772366067986465313t_bool(aTP_Lamm_gn(Uu),Uua),Uub))
    <=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s5460976970255530739at_nat(Uub)),Uua)) ) ) ).

% ATP.lambda_252
tff(fact_10375_ATP_Olambda__253,axiom,
    ! [Uu: set_Extended_enat,Uua: nat,Uub: list_Extended_enat] :
      ( pp(aa_lis1364384421681148007t_bool(aa_nat4017767907309459068t_bool(aTP_Lamm_go(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(set_Extended_enat2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s3941691890525107288d_enat(Uub)),Uua)) ) ) ).

% ATP.lambda_253
tff(fact_10376_ATP_Olambda__254,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_gm(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua)) ) ) ).

% ATP.lambda_254
tff(fact_10377_ATP_Olambda__255,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_gq(Uu),Uua),Uub))
    <=> ( pp(aa_set_bool_bool(ord_less_eq_set_bool(set_bool2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Uub)),Uua)) ) ) ).

% ATP.lambda_255
tff(fact_10378_ATP_Olambda__256,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_gr(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Uub)),Uua)) ) ) ).

% ATP.lambda_256
tff(fact_10379_ATP_Olambda__257,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_gs(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Uub)),Uua)) ) ) ).

% ATP.lambda_257
tff(fact_10380_ATP_Olambda__258,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_gg(Uu),Uua),Uub))
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_258
tff(fact_10381_ATP_Olambda__259,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( pp(aa_lis6649954313659033868t_bool(aa_nat1772366067986465313t_bool(aTP_Lamm_ge(Uu),Uua),Uub))
    <=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Uub)),Uu))
        & ( size_s5460976970255530739at_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_259
tff(fact_10382_ATP_Olambda__260,axiom,
    ! [Uu: set_Extended_enat,Uua: nat,Uub: list_Extended_enat] :
      ( pp(aa_lis1364384421681148007t_bool(aa_nat4017767907309459068t_bool(aTP_Lamm_gf(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(set_Extended_enat2(Uub)),Uu))
        & ( size_s3941691890525107288d_enat(Uub) = Uua ) ) ) ).

% ATP.lambda_260
tff(fact_10383_ATP_Olambda__261,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_gd(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_261
tff(fact_10384_ATP_Olambda__262,axiom,
    ! [Uu: set_list_nat,Uua: nat,Uub: list_list_nat] :
      ( pp(aa_lis4609933955469982953t_bool(aa_nat3401988846471794802t_bool(aTP_Lamm_gk(Uu),Uua),Uub))
    <=> ( ord_le6045566169113846134st_nat(set_list_nat2(Uub),Uu)
        & ( size_s3023201423986296836st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_262
tff(fact_10385_ATP_Olambda__263,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_gh(Uu),Uua),Uub))
    <=> ( pp(aa_set_bool_bool(ord_less_eq_set_bool(set_bool2(Uub)),Uu))
        & ( size_size_list_bool(Uub) = Uua ) ) ) ).

% ATP.lambda_263
tff(fact_10386_ATP_Olambda__264,axiom,
    ! [Uu: set_set_nat,Uua: nat,Uub: list_set_nat] :
      ( pp(aa_list_set_nat_bool(aa_nat948039967059030092t_bool(aTP_Lamm_gl(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(Uub)),Uu))
        & ( size_s3254054031482475050et_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_264
tff(fact_10387_ATP_Olambda__265,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_gi(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Uub)),Uu))
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_265
tff(fact_10388_ATP_Olambda__266,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_gj(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Uub)),Uu))
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_266
tff(fact_10389_ATP_Olambda__267,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_wq(Uu),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_267
tff(fact_10390_ATP_Olambda__268,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gb(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_268
tff(fact_10391_ATP_Olambda__269,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_nat8857836526276733660t_bool(aTP_Lamm_ui(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),pow_nat(set_or4665077453230672383an_nat(zero_zero_nat,Uu))))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_269
tff(fact_10392_ATP_Olambda__270,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_wx(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uub)) ) ) ).

% ATP.lambda_270
tff(fact_10393_ATP_Olambda__271,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ga(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),aa_nat_nat(suc,Uua))) ) ) ).

% ATP.lambda_271
tff(fact_10394_ATP_Olambda__272,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_sb(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uu),Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_272
tff(fact_10395_ATP_Olambda__273,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sd(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uu),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_273
tff(fact_10396_ATP_Olambda__274,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_tg(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uua),Uub)),aa_nat_complex(power_power_complex(Uu),Uub)) ).

% ATP.lambda_274
tff(fact_10397_ATP_Olambda__275,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_tf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uua),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_275
tff(fact_10398_ATP_Olambda__276,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_to(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_276
tff(fact_10399_ATP_Olambda__277,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: set_Pr8693737435421807431at_nat,Uub: produc859450856879609959at_nat] :
      ( pp(aa_Pro6098352494050694700t_bool(aa_set6997699984206591038t_bool(aTP_Lamm_ck(Uu),Uua),Uub))
    <=> ( member8206827879206165904at_nat(Uub,Uu)
        | member8206827879206165904at_nat(Uub,Uua) ) ) ).

% ATP.lambda_277
tff(fact_10400_ATP_Olambda__278,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: produc3843707927480180839at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(aa_set377272637606474814t_bool(aTP_Lamm_cl(Uu),Uua),Uub))
    <=> ( member8757157785044589968at_nat(Uub,Uu)
        | member8757157785044589968at_nat(Uub,Uua) ) ) ).

% ATP.lambda_278
tff(fact_10401_ATP_Olambda__279,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_cf(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        | pp(aa_set_real_bool(member_real(Uub),Uua)) ) ) ).

% ATP.lambda_279
tff(fact_10402_ATP_Olambda__280,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_set257547205302447420t_bool(aTP_Lamm_cg(Uu),Uua),Uub))
    <=> ( member_list_nat(Uub,Uu)
        | member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_280
tff(fact_10403_ATP_Olambda__281,axiom,
    ! [Uu: set_bool,Uua: set_bool,Uub: bool] :
      ( pp(aa_bool_bool(aa_set6670506438128111678l_bool(aTP_Lamm_ce(Uu),Uua),Uub))
    <=> ( pp(aa_set_bool_bool(member_bool(Uub),Uu))
        | pp(aa_set_bool_bool(member_bool(Uub),Uua)) ) ) ).

% ATP.lambda_281
tff(fact_10404_ATP_Olambda__282,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set9121396911425987952t_bool(aTP_Lamm_ch(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),Uu))
        | pp(aa_set_set_nat_bool(member_set_nat(Uub),Uua)) ) ) ).

% ATP.lambda_282
tff(fact_10405_ATP_Olambda__283,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_cj(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        | pp(aa_set_nat_bool(member_nat(Uub),Uua)) ) ) ).

% ATP.lambda_283
tff(fact_10406_ATP_Olambda__284,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_ci(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        | pp(aa_set_int_bool(member_int(Uub),Uua)) ) ) ).

% ATP.lambda_284
tff(fact_10407_ATP_Olambda__285,axiom,
    ! [Uu: product_prod_nat_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_set5223363028666124222t_bool(aTP_Lamm_ei(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Uub),Uua)) ) ) ).

% ATP.lambda_285
tff(fact_10408_ATP_Olambda__286,axiom,
    ! [Uu: real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_ek(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_real_bool(member_real(Uub),Uua)) ) ) ).

% ATP.lambda_286
tff(fact_10409_ATP_Olambda__287,axiom,
    ! [Uu: list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_set257547205302447420t_bool(aTP_Lamm_el(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_287
tff(fact_10410_ATP_Olambda__288,axiom,
    ! [Uu: bool,Uua: set_bool,Uub: bool] :
      ( pp(aa_bool_bool(aa_set6670506438128111678l_bool(aTP_Lamm_ej(Uu),Uua),Uub))
    <=> ( ( pp(Uub)
        <=> pp(Uu) )
        | pp(aa_set_bool_bool(member_bool(Uub),Uua)) ) ) ).

% ATP.lambda_288
tff(fact_10411_ATP_Olambda__289,axiom,
    ! [Uu: set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set9121396911425987952t_bool(aTP_Lamm_em(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_set_nat_bool(member_set_nat(Uub),Uua)) ) ) ).

% ATP.lambda_289
tff(fact_10412_ATP_Olambda__290,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_en(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_nat_bool(member_nat(Uub),Uua)) ) ) ).

% ATP.lambda_290
tff(fact_10413_ATP_Olambda__291,axiom,
    ! [Uu: int,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_eo(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_int_bool(member_int(Uub),Uua)) ) ) ).

% ATP.lambda_291
tff(fact_10414_ATP_Olambda__292,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ty(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uua))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_292
tff(fact_10415_ATP_Olambda__293,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_bm(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_293
tff(fact_10416_ATP_Olambda__294,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_tx(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_294
tff(fact_10417_ATP_Olambda__295,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_bo(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).

% ATP.lambda_295
tff(fact_10418_ATP_Olambda__296,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_bp(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_296
tff(fact_10419_ATP_Olambda__297,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_bn(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).

% ATP.lambda_297
tff(fact_10420_ATP_Olambda__298,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_av(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(vEBT_vebt_member(Uu),Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uub)) ) ) ).

% ATP.lambda_298
tff(fact_10421_ATP_Olambda__299,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_aw(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(vEBT_vebt_member(Uu),Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_299
tff(fact_10422_ATP_Olambda__300,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ww(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uua))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uu)) ) ) ).

% ATP.lambda_300
tff(fact_10423_ATP_Olambda__301,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_set5223363028666124222t_bool(aTP_Lamm_fu(Uu),Uua),Uub))
    <=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Uub),Uu))
        & pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Uub),Uua)) ) ) ).

% ATP.lambda_301
tff(fact_10424_ATP_Olambda__302,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_fp(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & pp(aa_set_real_bool(member_real(Uub),Uua)) ) ) ).

% ATP.lambda_302
tff(fact_10425_ATP_Olambda__303,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_set257547205302447420t_bool(aTP_Lamm_fq(Uu),Uua),Uub))
    <=> ( member_list_nat(Uub,Uu)
        & member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_303
tff(fact_10426_ATP_Olambda__304,axiom,
    ! [Uu: set_bool,Uua: set_bool,Uub: bool] :
      ( pp(aa_bool_bool(aa_set6670506438128111678l_bool(aTP_Lamm_fo(Uu),Uua),Uub))
    <=> ( pp(aa_set_bool_bool(member_bool(Uub),Uu))
        & pp(aa_set_bool_bool(member_bool(Uub),Uua)) ) ) ).

% ATP.lambda_304
tff(fact_10427_ATP_Olambda__305,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set9121396911425987952t_bool(aTP_Lamm_fr(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),Uu))
        & pp(aa_set_set_nat_bool(member_set_nat(Uub),Uua)) ) ) ).

% ATP.lambda_305
tff(fact_10428_ATP_Olambda__306,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_ft(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_set_nat_bool(member_nat(Uub),Uua)) ) ) ).

% ATP.lambda_306
tff(fact_10429_ATP_Olambda__307,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_fs(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & pp(aa_set_int_bool(member_int(Uub),Uua)) ) ) ).

% ATP.lambda_307
tff(fact_10430_ATP_Olambda__308,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun4759809662886658982t_bool(aTP_Lamm_mr(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uu))
        & pp(aa_Ext2772979874139974381t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_308
tff(fact_10431_ATP_Olambda__309,axiom,
    ! [Uu: set_complex,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_mp(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_309
tff(fact_10432_ATP_Olambda__310,axiom,
    ! [Uu: set_real,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_fe(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_310
tff(fact_10433_ATP_Olambda__311,axiom,
    ! [Uu: set_list_nat,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_ff(Uu),Uua),Uub))
    <=> ( member_list_nat(Uub,Uu)
        & pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_311
tff(fact_10434_ATP_Olambda__312,axiom,
    ! [Uu: set_bool,Uua: fun_bool_bool,Uub: bool] :
      ( pp(aa_bool_bool(aa_fun8098540714523660868l_bool(aTP_Lamm_fd(Uu),Uua),Uub))
    <=> ( pp(aa_set_bool_bool(member_bool(Uub),Uu))
        & pp(aa_bool_bool(Uua,Uub)) ) ) ).

% ATP.lambda_312
tff(fact_10435_ATP_Olambda__313,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_fg(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),Uu))
        & pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_313
tff(fact_10436_ATP_Olambda__314,axiom,
    ! [Uu: set_nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_fh(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_314
tff(fact_10437_ATP_Olambda__315,axiom,
    ! [Uu: set_int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_fi(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_315
tff(fact_10438_ATP_Olambda__316,axiom,
    ! [Uu: fun_Pr7757163693017731481t_bool,Uua: product_prod_nat_nat,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(aTP_Lamm_du(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_Pro2232559814647311250t_bool(Uu,Uub)) ) ) ).

% ATP.lambda_316
tff(fact_10439_ATP_Olambda__317,axiom,
    ! [Uu: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_dx(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_real_bool(Uu,Uub)) ) ) ).

% ATP.lambda_317
tff(fact_10440_ATP_Olambda__318,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aTP_Lamm_dv(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_list_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_318
tff(fact_10441_ATP_Olambda__319,axiom,
    ! [Uu: fun_bool_bool,Uua: bool,Uub: bool] :
      ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_dy(Uu),Uua),Uub))
    <=> ( ( pp(Uua)
        <=> pp(Uub) )
        & pp(aa_bool_bool(Uu,Uub)) ) ) ).

% ATP.lambda_319
tff(fact_10442_ATP_Olambda__320,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_dw(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_set_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_320
tff(fact_10443_ATP_Olambda__321,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_dz(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_321
tff(fact_10444_ATP_Olambda__322,axiom,
    ! [Uu: fun_int_bool,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ea(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_int_bool(Uu,Uub)) ) ) ).

% ATP.lambda_322
tff(fact_10445_ATP_Olambda__323,axiom,
    ! [Uu: fun_Pr7757163693017731481t_bool,Uua: product_prod_nat_nat,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(aTP_Lamm_dn(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_Pro2232559814647311250t_bool(Uu,Uub)) ) ) ).

% ATP.lambda_323
tff(fact_10446_ATP_Olambda__324,axiom,
    ! [Uu: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_dq(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_real_bool(Uu,Uub)) ) ) ).

% ATP.lambda_324
tff(fact_10447_ATP_Olambda__325,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aTP_Lamm_do(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_list_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_325
tff(fact_10448_ATP_Olambda__326,axiom,
    ! [Uu: fun_bool_bool,Uua: bool,Uub: bool] :
      ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_dr(Uu),Uua),Uub))
    <=> ( ( pp(Uub)
        <=> pp(Uua) )
        & pp(aa_bool_bool(Uu,Uub)) ) ) ).

% ATP.lambda_326
tff(fact_10449_ATP_Olambda__327,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_dp(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_set_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_327
tff(fact_10450_ATP_Olambda__328,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ds(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_328
tff(fact_10451_ATP_Olambda__329,axiom,
    ! [Uu: fun_int_bool,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_dt(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_int_bool(Uu,Uub)) ) ) ).

% ATP.lambda_329
tff(fact_10452_ATP_Olambda__330,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun338017137003533149t_bool(aTP_Lamm_jg(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uu))
        & ( aa_Ext2541572767299617110t_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_330
tff(fact_10453_ATP_Olambda__331,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_je(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_331
tff(fact_10454_ATP_Olambda__332,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_iy(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_332
tff(fact_10455_ATP_Olambda__333,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aTP_Lamm_jk(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_333
tff(fact_10456_ATP_Olambda__334,axiom,
    ! [Uu: set_bool,Uua: fun_bool_real,Uub: bool] :
      ( pp(aa_bool_bool(aa_fun2512541629346179229l_bool(aTP_Lamm_iw(Uu),Uua),Uub))
    <=> ( pp(aa_set_bool_bool(member_bool(Uub),Uu))
        & ( aa_bool_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_334
tff(fact_10457_ATP_Olambda__335,axiom,
    ! [Uu: set_bool,Uua: fun_bool_rat,Uub: bool] :
      ( pp(aa_bool_bool(aa_fun4510340986217973961l_bool(aTP_Lamm_ji(Uu),Uua),Uub))
    <=> ( pp(aa_set_bool_bool(member_bool(Uub),Uu))
        & ( aa_bool_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_335
tff(fact_10458_ATP_Olambda__336,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_ja(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_336
tff(fact_10459_ATP_Olambda__337,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aTP_Lamm_jm(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_337
tff(fact_10460_ATP_Olambda__338,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_jc(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_338
tff(fact_10461_ATP_Olambda__339,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: int] :
      ( pp(aa_int_bool(aa_fun258058769140594373t_bool(aTP_Lamm_jo(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_339
tff(fact_10462_ATP_Olambda__340,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex3741802288643809319omplex,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun7434105802671982559t_bool(aTP_Lamm_im(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uu))
        & ( aa_Ext8748583412541189208omplex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_340
tff(fact_10463_ATP_Olambda__341,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aTP_Lamm_ik(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_341
tff(fact_10464_ATP_Olambda__342,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aTP_Lamm_ie(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_342
tff(fact_10465_ATP_Olambda__343,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_iq(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_343
tff(fact_10466_ATP_Olambda__344,axiom,
    ! [Uu: set_bool,Uua: fun_bool_complex,Uub: bool] :
      ( pp(aa_bool_bool(aa_fun4000492451996535199l_bool(aTP_Lamm_ic(Uu),Uua),Uub))
    <=> ( pp(aa_set_bool_bool(member_bool(Uub),Uu))
        & ( aa_bool_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_344
tff(fact_10467_ATP_Olambda__345,axiom,
    ! [Uu: set_bool,Uua: fun_bool_real,Uub: bool] :
      ( pp(aa_bool_bool(aa_fun2512541629346179229l_bool(aTP_Lamm_io(Uu),Uua),Uub))
    <=> ( pp(aa_set_bool_bool(member_bool(Uub),Uu))
        & ( aa_bool_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_345
tff(fact_10468_ATP_Olambda__346,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aTP_Lamm_ig(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_346
tff(fact_10469_ATP_Olambda__347,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_is(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_347
tff(fact_10470_ATP_Olambda__348,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aTP_Lamm_ii(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_348
tff(fact_10471_ATP_Olambda__349,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_iu(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_349
tff(fact_10472_ATP_Olambda__350,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_bu(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ~ pp(aa_set_real_bool(member_real(Uub),Uua)) ) ) ).

% ATP.lambda_350
tff(fact_10473_ATP_Olambda__351,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_set257547205302447420t_bool(aTP_Lamm_bv(Uu),Uua),Uub))
    <=> ( member_list_nat(Uub,Uu)
        & ~ member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_351
tff(fact_10474_ATP_Olambda__352,axiom,
    ! [Uu: set_bool,Uua: set_bool,Uub: bool] :
      ( pp(aa_bool_bool(aa_set6670506438128111678l_bool(aTP_Lamm_bt(Uu),Uua),Uub))
    <=> ( pp(aa_set_bool_bool(member_bool(Uub),Uu))
        & ~ pp(aa_set_bool_bool(member_bool(Uub),Uua)) ) ) ).

% ATP.lambda_352
tff(fact_10475_ATP_Olambda__353,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set9121396911425987952t_bool(aTP_Lamm_bw(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),Uu))
        & ~ pp(aa_set_set_nat_bool(member_set_nat(Uub),Uua)) ) ) ).

% ATP.lambda_353
tff(fact_10476_ATP_Olambda__354,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_by(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ~ pp(aa_set_nat_bool(member_nat(Uub),Uua)) ) ) ).

% ATP.lambda_354
tff(fact_10477_ATP_Olambda__355,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_bx(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ~ pp(aa_set_int_bool(member_int(Uub),Uua)) ) ) ).

% ATP.lambda_355
tff(fact_10478_ATP_Olambda__356,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_vk(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uu)),Uua)) ) ).

% ATP.lambda_356
tff(fact_10479_ATP_Olambda__357,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_xh(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu)) ) ).

% ATP.lambda_357
tff(fact_10480_ATP_Olambda__358,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_wc(Uu),Uua),Uub))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu)) ) ).

% ATP.lambda_358
tff(fact_10481_ATP_Olambda__359,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_wa(Uu),Uua),Uub))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Uua),Uub)),Uu)) ) ).

% ATP.lambda_359
tff(fact_10482_ATP_Olambda__360,axiom,
    ! [Uu: set_Pr6588086440996610945on_nat,Uua: option_nat,Uub: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(aTP_Lamm_fj(Uu),Uua),Uub))
    <=> member4117937158525611210on_nat(produc5098337634421038937on_nat(Uua,Uub),Uu) ) ).

% ATP.lambda_360
tff(fact_10483_ATP_Olambda__361,axiom,
    ! [Uu: set_Pr7459493094073627847at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(aTP_Lamm_fk(Uu),Uua),Uub))
    <=> member1466754251312161552at_nat(produc9060074326276436823at_nat(Uua,Uub),Uu) ) ).

% ATP.lambda_361
tff(fact_10484_ATP_Olambda__362,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(aTP_Lamm_fl(Uu),Uua),Uub))
    <=> member8757157785044589968at_nat(produc2922128104949294807at_nat(Uua,Uub),Uu) ) ).

% ATP.lambda_362
tff(fact_10485_ATP_Olambda__363,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_fm(Uu),Uua),Uub))
    <=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)),Uu)) ) ).

% ATP.lambda_363
tff(fact_10486_ATP_Olambda__364,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_fn(Uu),Uua),Uub))
    <=> member5262025264175285858nt_int(product_Pair_int_int(Uua,Uub),Uu) ) ).

% ATP.lambda_364
tff(fact_10487_ATP_Olambda__365,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_ax(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_365
tff(fact_10488_ATP_Olambda__366,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_br(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_366
tff(fact_10489_ATP_Olambda__367,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_fw(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),ring_1_Ints_real))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uu),Uub))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uub),Uua)) ) ) ).

% ATP.lambda_367
tff(fact_10490_ATP_Olambda__368,axiom,
    ! [Uu: rat,Uua: rat,Uub: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_fx(Uu),Uua),Uub))
    <=> ( pp(aa_set_rat_bool(member_rat(Uub),ring_1_Ints_rat))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Uu),Uub))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Uub),Uua)) ) ) ).

% ATP.lambda_368
tff(fact_10491_ATP_Olambda__369,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_hx(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(suc,Uub))),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_369
tff(fact_10492_ATP_Olambda__370,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hy(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(suc,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_370
tff(fact_10493_ATP_Olambda__371,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vf(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_371
tff(fact_10494_ATP_Olambda__372,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_hq(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_372
tff(fact_10495_ATP_Olambda__373,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ho(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_373
tff(fact_10496_ATP_Olambda__374,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tp(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_374
tff(fact_10497_ATP_Olambda__375,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bq(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_375
tff(fact_10498_ATP_Olambda__376,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_wy(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_376
tff(fact_10499_ATP_Olambda__377,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ug(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_377
tff(fact_10500_ATP_Olambda__378,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_ph(Uu),Uua),Uub) = times_3573771949741848930nteger(aa_real_Code_integer(Uua,Uub),aa_real_Code_integer(Uu,Uub)) ).

% ATP.lambda_378
tff(fact_10501_ATP_Olambda__379,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_pl(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_379
tff(fact_10502_ATP_Olambda__380,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_po(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(Uua,Uub)),aa_real_rat(Uu,Uub)) ).

% ATP.lambda_380
tff(fact_10503_ATP_Olambda__381,axiom,
    ! [Uu: fun_bo7282069421189217543nteger,Uua: fun_bo7282069421189217543nteger,Uub: bool] : aa_bool_Code_integer(aa_fun1377107737837666940nteger(aTP_Lamm_pi(Uu),Uua),Uub) = times_3573771949741848930nteger(aa_bool_Code_integer(Uua,Uub),aa_bool_Code_integer(Uu,Uub)) ).

% ATP.lambda_381
tff(fact_10504_ATP_Olambda__382,axiom,
    ! [Uu: fun_bool_real,Uua: fun_bool_real,Uub: bool] : aa_bool_real(aa_fun8933258245593490108l_real(aTP_Lamm_pm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_bool_real(Uua,Uub)),aa_bool_real(Uu,Uub)) ).

% ATP.lambda_382
tff(fact_10505_ATP_Olambda__383,axiom,
    ! [Uu: fun_bool_rat,Uua: fun_bool_rat,Uub: bool] : aa_bool_rat(aa_fun224000846028461884ol_rat(aTP_Lamm_pp(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_bool_rat(Uua,Uub)),aa_bool_rat(Uu,Uub)) ).

% ATP.lambda_383
tff(fact_10506_ATP_Olambda__384,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_pj(Uu),Uua),Uub) = times_3573771949741848930nteger(aa_nat_Code_integer(Uua,Uub),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_384
tff(fact_10507_ATP_Olambda__385,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_pq(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_385
tff(fact_10508_ATP_Olambda__386,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_pk(Uu),Uua),Uub) = times_3573771949741848930nteger(aa_int_Code_integer(Uua,Uub),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_386
tff(fact_10509_ATP_Olambda__387,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_pn(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uub)),aa_int_real(Uu,Uub)) ).

% ATP.lambda_387
tff(fact_10510_ATP_Olambda__388,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vn(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_388
tff(fact_10511_ATP_Olambda__389,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_rm(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_real_nat(Uua,Uub)),aa_real_nat(Uu,Uub)) ).

% ATP.lambda_389
tff(fact_10512_ATP_Olambda__390,axiom,
    ! [Uu: fun_bool_nat,Uua: fun_bool_nat,Uub: bool] : aa_bool_nat(aa_fun9101744387563128124ol_nat(aTP_Lamm_rn(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_bool_nat(Uua,Uub)),aa_bool_nat(Uu,Uub)) ).

% ATP.lambda_390
tff(fact_10513_ATP_Olambda__391,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_ro(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(Uua,Uub)),aa_set_nat_nat(Uu,Uub)) ).

% ATP.lambda_391
tff(fact_10514_ATP_Olambda__392,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_my(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_392
tff(fact_10515_ATP_Olambda__393,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rq(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uua,Uub)),aa_nat_nat(Uu,Uub)) ).

% ATP.lambda_393
tff(fact_10516_ATP_Olambda__394,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_rp(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uua,Uub)),aa_int_nat(Uu,Uub)) ).

% ATP.lambda_394
tff(fact_10517_ATP_Olambda__395,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_uw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_395
tff(fact_10518_ATP_Olambda__396,axiom,
    ! [Uu: fun_Pr446604468264085555t_bool,Uua: fun_Pr446604468264085555t_bool,Uub: produc859450856879609959at_nat] :
      ( pp(aa_Pro6098352494050694700t_bool(aa_fun3665199231186260388t_bool(aTP_Lamm_fa(Uu),Uua),Uub))
    <=> ( pp(aa_Pro6098352494050694700t_bool(Uu,Uub))
       => pp(aa_Pro6098352494050694700t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_396
tff(fact_10519_ATP_Olambda__397,axiom,
    ! [Uu: fun_Pr1607443439640623155t_bool,Uua: fun_Pr1607443439640623155t_bool,Uub: produc3843707927480180839at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(aa_fun5022828243006652836t_bool(aTP_Lamm_fb(Uu),Uua),Uub))
    <=> ( pp(aa_Pro1809015098595223084t_bool(Uu,Uub))
       => pp(aa_Pro1809015098595223084t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_397
tff(fact_10520_ATP_Olambda__398,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_ev(Uu),Uua),Uub))
    <=> ( pp(aa_real_bool(Uu,Uub))
       => pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_398
tff(fact_10521_ATP_Olambda__399,axiom,
    ! [Uu: fun_list_nat_bool,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_ew(Uu),Uua),Uub))
    <=> ( pp(aa_list_nat_bool(Uu,Uub))
       => pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_399
tff(fact_10522_ATP_Olambda__400,axiom,
    ! [Uu: fun_set_nat_bool,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_ex(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(Uu,Uub))
       => pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_400
tff(fact_10523_ATP_Olambda__401,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_ez(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
       => pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_401
tff(fact_10524_ATP_Olambda__402,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_ey(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(Uu,Uub))
       => pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_402
tff(fact_10525_ATP_Olambda__403,axiom,
    ! [Uu: fun_Pr446604468264085555t_bool,Uua: fun_Pr446604468264085555t_bool,Uub: produc859450856879609959at_nat] :
      ( pp(aa_Pro6098352494050694700t_bool(aa_fun3665199231186260388t_bool(aTP_Lamm_co(Uu),Uua),Uub))
    <=> ( pp(aa_Pro6098352494050694700t_bool(Uu,Uub))
        | pp(aa_Pro6098352494050694700t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_403
tff(fact_10526_ATP_Olambda__404,axiom,
    ! [Uu: fun_Pr1607443439640623155t_bool,Uua: fun_Pr1607443439640623155t_bool,Uub: produc3843707927480180839at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(aa_fun5022828243006652836t_bool(aTP_Lamm_cp(Uu),Uua),Uub))
    <=> ( pp(aa_Pro1809015098595223084t_bool(Uu,Uub))
        | pp(aa_Pro1809015098595223084t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_404
tff(fact_10527_ATP_Olambda__405,axiom,
    ! [Uu: fun_Pr7757163693017731481t_bool,Uua: fun_Pr7757163693017731481t_bool,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_fun830836278467416944t_bool(aTP_Lamm_at(Uu),Uua),Uub))
    <=> ( pp(aa_Pro2232559814647311250t_bool(Uu,Uub))
        | pp(aa_Pro2232559814647311250t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_405
tff(fact_10528_ATP_Olambda__406,axiom,
    ! [Uu: fun_Ex8563670519792538868t_bool,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun4759809662886658982t_bool(aTP_Lamm_au(Uu),Uua),Uub))
    <=> ( pp(aa_Ext2772979874139974381t_bool(Uu,Uub))
        | pp(aa_Ext2772979874139974381t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_406
tff(fact_10529_ATP_Olambda__407,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_as(Uu),Uua),Uub))
    <=> ( pp(aa_complex_bool(Uu,Uub))
        | pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_407
tff(fact_10530_ATP_Olambda__408,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_an(Uu),Uua),Uub))
    <=> ( pp(aa_real_bool(Uu,Uub))
        | pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_408
tff(fact_10531_ATP_Olambda__409,axiom,
    ! [Uu: fun_list_nat_bool,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_ao(Uu),Uua),Uub))
    <=> ( pp(aa_list_nat_bool(Uu,Uub))
        | pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_409
tff(fact_10532_ATP_Olambda__410,axiom,
    ! [Uu: fun_set_nat_bool,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_ap(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(Uu,Uub))
        | pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_410
tff(fact_10533_ATP_Olambda__411,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_aq(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        | pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_411
tff(fact_10534_ATP_Olambda__412,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_ar(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(Uu,Uub))
        | pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_412
tff(fact_10535_ATP_Olambda__413,axiom,
    ! [Uu: fun_Pr7757163693017731481t_bool,Uua: fun_Pr7757163693017731481t_bool,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_fun830836278467416944t_bool(aTP_Lamm_al(Uu),Uua),Uub))
    <=> ( pp(aa_Pro2232559814647311250t_bool(Uu,Uub))
        & pp(aa_Pro2232559814647311250t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_413
tff(fact_10536_ATP_Olambda__414,axiom,
    ! [Uu: fun_Ex8563670519792538868t_bool,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun4759809662886658982t_bool(aTP_Lamm_am(Uu),Uua),Uub))
    <=> ( pp(aa_Ext2772979874139974381t_bool(Uu,Uub))
        & pp(aa_Ext2772979874139974381t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_414
tff(fact_10537_ATP_Olambda__415,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_ak(Uu),Uua),Uub))
    <=> ( pp(aa_complex_bool(Uu,Uub))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_415
tff(fact_10538_ATP_Olambda__416,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_af(Uu),Uua),Uub))
    <=> ( pp(aa_real_bool(Uu,Uub))
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_416
tff(fact_10539_ATP_Olambda__417,axiom,
    ! [Uu: fun_list_nat_bool,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_ag(Uu),Uua),Uub))
    <=> ( pp(aa_list_nat_bool(Uu,Uub))
        & pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_417
tff(fact_10540_ATP_Olambda__418,axiom,
    ! [Uu: fun_set_nat_bool,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_ah(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(Uu,Uub))
        & pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_418
tff(fact_10541_ATP_Olambda__419,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_ai(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_419
tff(fact_10542_ATP_Olambda__420,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_aj(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(Uu,Uub))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_420
tff(fact_10543_ATP_Olambda__421,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_fc(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua)) ) ).

% ATP.lambda_421
tff(fact_10544_ATP_Olambda__422,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_hg(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_422
tff(fact_10545_ATP_Olambda__423,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_423
tff(fact_10546_ATP_Olambda__424,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_na(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uub)),Uu) ).

% ATP.lambda_424
tff(fact_10547_ATP_Olambda__425,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_nb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_425
tff(fact_10548_ATP_Olambda__426,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vu(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_426
tff(fact_10549_ATP_Olambda__427,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_uv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_427
tff(fact_10550_ATP_Olambda__428,axiom,
    ! [Uu: product_prod_nat_nat,Uua: fun_Pr7757163693017731481t_bool,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_fun830836278467416944t_bool(aTP_Lamm_eb(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_Pro2232559814647311250t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_428
tff(fact_10551_ATP_Olambda__429,axiom,
    ! [Uu: real,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_ed(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_429
tff(fact_10552_ATP_Olambda__430,axiom,
    ! [Uu: list_nat,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_ee(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_430
tff(fact_10553_ATP_Olambda__431,axiom,
    ! [Uu: bool,Uua: fun_bool_bool,Uub: bool] :
      ( pp(aa_bool_bool(aa_fun8098540714523660868l_bool(aTP_Lamm_ec(Uu),Uua),Uub))
    <=> ( ~ ( pp(Uub)
          <=> pp(Uu) )
       => pp(aa_bool_bool(Uua,Uub)) ) ) ).

% ATP.lambda_431
tff(fact_10554_ATP_Olambda__432,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_ef(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_432
tff(fact_10555_ATP_Olambda__433,axiom,
    ! [Uu: nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_eg(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_433
tff(fact_10556_ATP_Olambda__434,axiom,
    ! [Uu: int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_eh(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_434
tff(fact_10557_ATP_Olambda__435,axiom,
    ! [Uu: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_ib(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uua,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_435
tff(fact_10558_ATP_Olambda__436,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_qi(Uu),Uua),Uub) = plus_p5714425477246183910nteger(Uu,times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Uub),Uua)) ).

% ATP.lambda_436
tff(fact_10559_ATP_Olambda__437,axiom,
    ! [Uu: extended_enat,Uua: extended_enat,Uub: nat] : aa_nat_Extended_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_qh(Uu),Uua),Uub) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Uu),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uub)),Uua)) ).

% ATP.lambda_437
tff(fact_10560_ATP_Olambda__438,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_qf(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_438
tff(fact_10561_ATP_Olambda__439,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ql(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_439
tff(fact_10562_ATP_Olambda__440,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_qg(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_440
tff(fact_10563_ATP_Olambda__441,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qk(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Uub)),Uua)) ).

% ATP.lambda_441
tff(fact_10564_ATP_Olambda__442,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_qj(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Uub)),Uua)) ).

% ATP.lambda_442
tff(fact_10565_ATP_Olambda__443,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_tb(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_443
tff(fact_10566_ATP_Olambda__444,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_te(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_444
tff(fact_10567_ATP_Olambda__445,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_tc(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_445
tff(fact_10568_ATP_Olambda__446,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_td(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_446
tff(fact_10569_ATP_Olambda__447,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vq(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_447
tff(fact_10570_ATP_Olambda__448,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ok(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_448
tff(fact_10571_ATP_Olambda__449,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_on(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_449
tff(fact_10572_ATP_Olambda__450,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_ol(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_450
tff(fact_10573_ATP_Olambda__451,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_om(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_451
tff(fact_10574_ATP_Olambda__452,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qm(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_452
tff(fact_10575_ATP_Olambda__453,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_vw(Uu),Uua),Uub))
    <=> pp(aa_set_real_bool(member_real(Uub),set_or1633881224788618240n_real(Uu,Uua))) ) ).

% ATP.lambda_453
tff(fact_10576_ATP_Olambda__454,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hd(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_454
tff(fact_10577_ATP_Olambda__455,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_he(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_455
tff(fact_10578_ATP_Olambda__456,axiom,
    ! [Uu: real,Uua: real,Uub: product_unit] : aa_Product_unit_real(aa_rea7683167438103758690t_real(aTP_Lamm_xn(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real2,Uu),Uua) ).

% ATP.lambda_456
tff(fact_10579_ATP_Olambda__457,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_sl(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_457
tff(fact_10580_ATP_Olambda__458,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sk(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_458
tff(fact_10581_ATP_Olambda__459,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ia(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_459
tff(fact_10582_ATP_Olambda__460,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_hz(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_460
tff(fact_10583_ATP_Olambda__461,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nx(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_461
tff(fact_10584_ATP_Olambda__462,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_nz(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_462
tff(fact_10585_ATP_Olambda__463,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_wi(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_463
tff(fact_10586_ATP_Olambda__464,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_hr(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub))) ).

% ATP.lambda_464
tff(fact_10587_ATP_Olambda__465,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hp(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub))) ).

% ATP.lambda_465
tff(fact_10588_ATP_Olambda__466,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_re7357811994743134493t_bool,Uub: real] : aa_real_nat(aa_fun6765506734994765012al_nat(aTP_Lamm_rl(Uu),Uua),Uub) = finite121521170596916366d_enat(collec4429806609662206161d_enat(aa_rea3571252896447469150t_bool(aa_fun6444895427502343800t_bool(aTP_Lamm_rk(Uu),Uua),Uub))) ).

% ATP.lambda_466
tff(fact_10589_ATP_Olambda__467,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_bo6623066019567229854t_bool,Uub: bool] : aa_bool_nat(aa_fun85076598157486616ol_nat(aTP_Lamm_rj(Uu),Uua),Uub) = finite121521170596916366d_enat(collec4429806609662206161d_enat(aa_boo1364755444788961487t_bool(aa_fun5649173921161459132t_bool(aTP_Lamm_ri(Uu),Uua),Uub))) ).

% ATP.lambda_467
tff(fact_10590_ATP_Olambda__468,axiom,
    ! [Uu: set_int,Uua: fun_re8661203372455305201t_bool,Uub: real] : aa_real_nat(aa_fun4707068664741690670al_nat(aTP_Lamm_rf(Uu),Uua),Uub) = finite_card_int(collect_int(aa_real_fun_int_bool(aa_fun6227302519962330364t_bool(aTP_Lamm_re(Uu),Uua),Uub))) ).

% ATP.lambda_468
tff(fact_10591_ATP_Olambda__469,axiom,
    ! [Uu: set_int,Uua: fun_bo2827871560234191408t_bool,Uub: bool] : aa_bool_nat(aa_fun7155190969624455820ol_nat(aTP_Lamm_rd(Uu),Uua),Uub) = finite_card_int(collect_int(aa_bool_fun_int_bool(aa_fun4998116011708744478t_bool(aTP_Lamm_rc(Uu),Uua),Uub))) ).

% ATP.lambda_469
tff(fact_10592_ATP_Olambda__470,axiom,
    ! [Uu: set_real,Uua: fun_re1255401033344146673l_bool,Uub: real] : aa_real_nat(aa_fun3074920444450379950al_nat(aTP_Lamm_rb(Uu),Uua),Uub) = finite_card_real(collect_real(aa_rea7809428808904475482l_bool(aa_fun1425803159125577980l_bool(aTP_Lamm_ra(Uu),Uua),Uub))) ).

% ATP.lambda_470
tff(fact_10593_ATP_Olambda__471,axiom,
    ! [Uu: set_bool,Uua: fun_re3316061561565328172l_bool,Uub: real] : aa_real_nat(aa_fun4868894756110751843al_nat(aTP_Lamm_qy(Uu),Uua),Uub) = finite_card_bool(collect_bool(aa_rea6668032140610767853l_bool(aa_fun1775789228971148182l_bool(aTP_Lamm_qx(Uu),Uua),Uub))) ).

% ATP.lambda_471
tff(fact_10594_ATP_Olambda__472,axiom,
    ! [Uu: set_real,Uua: fun_bo2025495424286513072l_bool,Uub: bool] : aa_bool_nat(aa_fun2932017502249704332ol_nat(aTP_Lamm_qv(Uu),Uua),Uub) = finite_card_real(collect_real(aa_boo2589871674041770537l_bool(aa_fun1574454591505385630l_bool(aTP_Lamm_qu(Uu),Uua),Uub))) ).

% ATP.lambda_472
tff(fact_10595_ATP_Olambda__473,axiom,
    ! [Uu: set_bool,Uua: fun_bo1841576699660343469l_bool,Uub: bool] : aa_bool_nat(aa_fun3941158123937086921ol_nat(aTP_Lamm_qs(Uu),Uua),Uub) = finite_card_bool(collect_bool(aa_boo8970175072276269406l_bool(aa_fun1447330757145981436l_bool(aTP_Lamm_qr(Uu),Uua),Uub))) ).

% ATP.lambda_473
tff(fact_10596_ATP_Olambda__474,axiom,
    ! [Uu: set_complex,Uua: fun_re3010880487650781423x_bool,Uub: real] : aa_real_nat(aa_fun5167920629197714732al_nat(aTP_Lamm_rh(Uu),Uua),Uub) = finite_card_complex(collect_complex(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_ls(Uu),Uua),Uub))) ).

% ATP.lambda_474
tff(fact_10597_ATP_Olambda__475,axiom,
    ! [Uu: set_complex,Uua: fun_bo8977660800532043054x_bool,Uub: bool] : aa_bool_nat(aa_fun8646024951575708942ol_nat(aTP_Lamm_rg(Uu),Uua),Uub) = finite_card_complex(collect_complex(aa_boo8509281623719168167x_bool(aa_fun8473158556963620890x_bool(aTP_Lamm_lo(Uu),Uua),Uub))) ).

% ATP.lambda_475
tff(fact_10598_ATP_Olambda__476,axiom,
    ! [Uu: set_nat,Uua: fun_re9121486324101951821t_bool,Uub: real] : aa_real_nat(aa_fun5804151645385689226al_nat(aTP_Lamm_rs(Uu),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_kv(Uu),Uua),Uub))) ).

% ATP.lambda_476
tff(fact_10599_ATP_Olambda__477,axiom,
    ! [Uu: set_nat,Uua: fun_bo3288154511880838028t_bool,Uub: bool] : aa_bool_nat(aa_fun7928520491197679664ol_nat(aTP_Lamm_rr(Uu),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_bool_fun_nat_bool(aa_fun7537470554201192406t_bool(aTP_Lamm_kr(Uu),Uua),Uub))) ).

% ATP.lambda_477
tff(fact_10600_ATP_Olambda__478,axiom,
    ! [Uu: fun_nat_real,Uua: set_nat,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_mz(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_set_nat_bool(member_nat(Uuc),Uua)),aa_nat_real(Uub,Uuc)),aa_nat_real(Uu,Uuc)) ).

% ATP.lambda_478
tff(fact_10601_ATP_Olambda__479,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: fun_Ex1451620480498224165t_real,Uuc: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1749635019461158908t_real(aa_fun967516747568145260t_real(aTP_Lamm_or(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uuc),Uu)),aa_Ext2541572767299617110t_real(Uua,Uuc)),aa_Ext2541572767299617110t_real(Uub,Uuc)) ).

% ATP.lambda_479
tff(fact_10602_ATP_Olambda__480,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex4152069839811072081at_rat,Uub: fun_Ex4152069839811072081at_rat,Uuc: extended_enat] : aa_Extended_enat_rat(aa_fun9057213682095532284at_rat(aa_fun4607285249076117784at_rat(aTP_Lamm_ot(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uuc),Uu)),aa_Extended_enat_rat(Uua,Uuc)),aa_Extended_enat_rat(Uub,Uuc)) ).

% ATP.lambda_480
tff(fact_10603_ATP_Olambda__481,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex7660298398204468809at_nat,Uub: fun_Ex7660298398204468809at_nat,Uuc: extended_enat] : aa_Extended_enat_nat(aa_fun1810565495048290044at_nat(aa_fun5812849712608808208at_nat(aTP_Lamm_ov(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uuc),Uu)),aa_Extended_enat_nat(Uua,Uuc)),aa_Extended_enat_nat(Uub,Uuc)) ).

% ATP.lambda_481
tff(fact_10604_ATP_Olambda__482,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex3482447378695272101at_int,Uub: fun_Ex3482447378695272101at_int,Uuc: extended_enat] : aa_Extended_enat_int(aa_fun1695976788104708604at_int(aa_fun8556764835550362732at_int(aTP_Lamm_ox(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(if_int(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,Uuc),Uu)),aa_Extended_enat_int(Uua,Uuc)),aa_Extended_enat_int(Uub,Uuc)) ).

% ATP.lambda_482
tff(fact_10605_ATP_Olambda__483,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_oq(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu)),aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_483
tff(fact_10606_ATP_Olambda__484,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_os(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu)),aa_complex_rat(Uua,Uuc)),aa_complex_rat(Uub,Uuc)) ).

% ATP.lambda_484
tff(fact_10607_ATP_Olambda__485,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_ou(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu)),aa_complex_nat(Uua,Uuc)),aa_complex_nat(Uub,Uuc)) ).

% ATP.lambda_485
tff(fact_10608_ATP_Olambda__486,axiom,
    ! [Uu: complex,Uua: fun_complex_int,Uub: fun_complex_int,Uuc: complex] : aa_complex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_ow(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(if_int(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu)),aa_complex_int(Uua,Uuc)),aa_complex_int(Uub,Uuc)) ).

% ATP.lambda_486
tff(fact_10609_ATP_Olambda__487,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] : aa_real_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_oy(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu)),aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) ).

% ATP.lambda_487
tff(fact_10610_ATP_Olambda__488,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_oz(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu)),aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) ).

% ATP.lambda_488
tff(fact_10611_ATP_Olambda__489,axiom,
    ! [Uu: fun_Ex8563670519792538868t_bool,Uua: fun_Ex1451620480498224165t_real,Uub: fun_Ex1451620480498224165t_real,Uuc: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1749635019461158908t_real(aa_fun967516747568145260t_real(aTP_Lamm_od(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Ext2772979874139974381t_bool(Uu,Uuc)),aa_Ext2541572767299617110t_real(Uua,Uuc)),aa_Ext2541572767299617110t_real(Uub,Uuc)) ).

% ATP.lambda_489
tff(fact_10612_ATP_Olambda__490,axiom,
    ! [Uu: fun_Ex8563670519792538868t_bool,Uua: fun_Ex4152069839811072081at_rat,Uub: fun_Ex4152069839811072081at_rat,Uuc: extended_enat] : aa_Extended_enat_rat(aa_fun9057213682095532284at_rat(aa_fun4607285249076117784at_rat(aTP_Lamm_oh(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_Ext2772979874139974381t_bool(Uu,Uuc)),aa_Extended_enat_rat(Uua,Uuc)),aa_Extended_enat_rat(Uub,Uuc)) ).

% ATP.lambda_490
tff(fact_10613_ATP_Olambda__491,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_oc(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(Uu,Uuc)),aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_491
tff(fact_10614_ATP_Olambda__492,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_og(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_complex_bool(Uu,Uuc)),aa_complex_rat(Uua,Uuc)),aa_complex_rat(Uub,Uuc)) ).

% ATP.lambda_492
tff(fact_10615_ATP_Olambda__493,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] : aa_real_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_oa(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(Uu,Uuc)),aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) ).

% ATP.lambda_493
tff(fact_10616_ATP_Olambda__494,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_oe(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_real_bool(Uu,Uuc)),aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) ).

% ATP.lambda_494
tff(fact_10617_ATP_Olambda__495,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_nat,Uub: fun_real_nat,Uuc: real] : aa_real_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_oi(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(Uu,Uuc)),aa_real_nat(Uua,Uuc)),aa_real_nat(Uub,Uuc)) ).

% ATP.lambda_495
tff(fact_10618_ATP_Olambda__496,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] : aa_int_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_ob(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(Uu,Uuc)),aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) ).

% ATP.lambda_496
tff(fact_10619_ATP_Olambda__497,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_of(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_int_bool(Uu,Uuc)),aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) ).

% ATP.lambda_497
tff(fact_10620_ATP_Olambda__498,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_oj(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(Uu,Uuc)),aa_int_nat(Uua,Uuc)),aa_int_nat(Uub,Uuc)) ).

% ATP.lambda_498
tff(fact_10621_ATP_Olambda__499,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_sa(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_rz(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_499
tff(fact_10622_ATP_Olambda__500,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ry(Uu),Uua),Uub),Uuc) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_rx(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_500
tff(fact_10623_ATP_Olambda__501,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rw(Uu),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rv(Uu),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_501
tff(fact_10624_ATP_Olambda__502,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex6753287174829011282omplex,Uub: fun_Ex650070010719229343x_bool,Uuc: complex] : aa_complex_complex(aa_fun2004122418815753142omplex(aa_fun1910344193496359307omplex(aTP_Lamm_mk(Uu),Uua),Uub),Uuc) = groups6818542070133387226omplex(aa_com6527309619268954171omplex(aTP_Lamm_mi(Uua),Uuc),collec4429806609662206161d_enat(aa_com1925010942101449184t_bool(aa_fun1028883470959682428t_bool(aTP_Lamm_mj(Uu),Uub),Uuc))) ).

% ATP.lambda_502
tff(fact_10625_ATP_Olambda__503,axiom,
    ! [Uu: set_int,Uua: fun_in1307110748760623106omplex,Uub: fun_in5750378413879446255x_bool,Uuc: complex] : aa_complex_complex(aa_fun2686559102040213132omplex(aa_fun5422379709423809691omplex(aTP_Lamm_mf(Uu),Uua),Uub),Uuc) = groups3049146728041665814omplex(aa_com9156536935073639875omplex(aTP_Lamm_md(Uua),Uuc),collect_int(aa_com2211550117870696024t_bool(aa_fun3087709329828230012t_bool(aTP_Lamm_me(Uu),Uub),Uuc))) ).

% ATP.lambda_503
tff(fact_10626_ATP_Olambda__504,axiom,
    ! [Uu: set_nat,Uua: fun_na1591161823130732966omplex,Uub: fun_na5941084329111710283x_bool,Uuc: complex] : aa_complex_complex(aa_fun1613615970879741416omplex(aa_fun7586814990830484307omplex(aTP_Lamm_ma(Uu),Uua),Uub),Uuc) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ly(Uua),Uuc)),collect_nat(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_lz(Uu),Uub),Uuc))) ).

% ATP.lambda_504
tff(fact_10627_ATP_Olambda__505,axiom,
    ! [Uu: set_real,Uua: fun_re7208826350674946562omplex,Uub: fun_re3010880487650781423x_bool,Uuc: complex] : aa_complex_complex(aa_fun8128191506076533516omplex(aa_fun840110154952133019omplex(aTP_Lamm_lv(Uu),Uua),Uub),Uuc) = groups5754745047067104278omplex(aa_com4766338864777664963omplex(aTP_Lamm_lu(Uua),Uuc),collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_di(Uu),Uub),Uuc))) ).

% ATP.lambda_505
tff(fact_10628_ATP_Olambda__506,axiom,
    ! [Uu: set_bool,Uua: fun_bo6990880009230973955omplex,Uub: fun_bo8977660800532043054x_bool,Uuc: complex] : aa_complex_complex(aa_fun8239209723672002629omplex(aa_fun6558688929985837387omplex(aTP_Lamm_lr(Uu),Uua),Uub),Uuc) = groups2959343415743557771omplex(aa_com2616170237055825644omplex(aTP_Lamm_lq(Uua),Uuc),collect_bool(aa_com4332222969190720879l_bool(aa_fun6951469949174074266l_bool(aTP_Lamm_dh(Uu),Uub),Uuc))) ).

% ATP.lambda_506
tff(fact_10629_ATP_Olambda__507,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex7123704652682714774at_nat,Uub: fun_Ex7682446853002821117t_bool,Uuc: nat] : aa_nat_nat(aa_fun1897775202592251224at_nat(aa_fun594951741515554929at_nat(aTP_Lamm_ln(Uu),Uua),Uub),Uuc) = groups2027974829824023292at_nat(aa_nat887872048497836987at_nat(aTP_Lamm_ll(Uua),Uuc),collec4429806609662206161d_enat(aa_nat6328455298437991426t_bool(aa_fun955218191390214396t_bool(aTP_Lamm_lm(Uu),Uub),Uuc))) ).

% ATP.lambda_507
tff(fact_10630_ATP_Olambda__508,axiom,
    ! [Uu: set_complex,Uua: fun_co636942507509095880at_nat,Uub: fun_co2841405903858895691t_bool,Uuc: nat] : aa_nat_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_li(Uu),Uua),Uub),Uuc) = groups5693394587270226106ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_lg(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_lh(Uu),Uub),Uuc))) ).

% ATP.lambda_508
tff(fact_10631_ATP_Olambda__509,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_bool,Uuc: nat] : aa_nat_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_ld(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_lb(Uua),Uuc),collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_lc(Uu),Uub),Uuc))) ).

% ATP.lambda_509
tff(fact_10632_ATP_Olambda__510,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_re9121486324101951821t_bool,Uuc: nat] : aa_nat_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_ky(Uu),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(aTP_Lamm_kx(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_de(Uu),Uub),Uuc))) ).

% ATP.lambda_510
tff(fact_10633_ATP_Olambda__511,axiom,
    ! [Uu: set_bool,Uua: fun_bool_fun_nat_nat,Uub: fun_bo3288154511880838028t_bool,Uuc: nat] : aa_nat_nat(aa_fun4984880352635657191at_nat(aa_fun8028557108643889at_nat(aTP_Lamm_ku(Uu),Uua),Uub),Uuc) = groups5768501273728744877ol_nat(aa_nat_fun_bool_nat(aTP_Lamm_kt(Uua),Uuc),collect_bool(aa_nat_fun_bool_bool(aa_fun4163026448321506586l_bool(aTP_Lamm_dd(Uu),Uub),Uuc))) ).

% ATP.lambda_511
tff(fact_10634_ATP_Olambda__512,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_tr(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_tq(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_512
tff(fact_10635_ATP_Olambda__513,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_vd(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_513
tff(fact_10636_ATP_Olambda__514,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_vb(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_514
tff(fact_10637_ATP_Olambda__515,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_uz(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_515
tff(fact_10638_ATP_Olambda__516,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_va(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_516
tff(fact_10639_ATP_Olambda__517,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_rx(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uua),Uu)),Uuc)),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_517
tff(fact_10640_ATP_Olambda__518,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_rz(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu)),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_518
tff(fact_10641_ATP_Olambda__519,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rv(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uua),Uu)),Uuc)),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_519
tff(fact_10642_ATP_Olambda__520,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_st(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_complex(power_power_complex(Uu),Uuc)) ).

% ATP.lambda_520
tff(fact_10643_ATP_Olambda__521,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_sw(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)) ).

% ATP.lambda_521
tff(fact_10644_ATP_Olambda__522,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_su(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_rat(power_power_rat(Uu),Uuc)) ).

% ATP.lambda_522
tff(fact_10645_ATP_Olambda__523,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_sv(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_int(power_power_int(Uu),Uuc)) ).

% ATP.lambda_523
tff(fact_10646_ATP_Olambda__524,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_re7357811994743134493t_bool,Uub: real,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_rea3571252896447469150t_bool(aa_fun6444895427502343800t_bool(aTP_Lamm_rk(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu))
        & pp(aa_Ext2772979874139974381t_bool(aa_rea3571252896447469150t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_524
tff(fact_10647_ATP_Olambda__525,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_bo6623066019567229854t_bool,Uub: bool,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_boo1364755444788961487t_bool(aa_fun5649173921161459132t_bool(aTP_Lamm_ri(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu))
        & pp(aa_Ext2772979874139974381t_bool(aa_boo1364755444788961487t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_525
tff(fact_10648_ATP_Olambda__526,axiom,
    ! [Uu: set_complex,Uua: fun_Ex650070010719229343x_bool,Uub: extended_enat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_Ext6408740497486688536x_bool(aa_fun5941807508050755708x_bool(aTP_Lamm_mg(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_Ext6408740497486688536x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_526
tff(fact_10649_ATP_Olambda__527,axiom,
    ! [Uu: set_complex,Uua: fun_re3010880487650781423x_bool,Uub: real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_ls(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_527
tff(fact_10650_ATP_Olambda__528,axiom,
    ! [Uu: set_complex,Uua: fun_bo8977660800532043054x_bool,Uub: bool,Uuc: complex] :
      ( pp(aa_complex_bool(aa_boo8509281623719168167x_bool(aa_fun8473158556963620890x_bool(aTP_Lamm_lo(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_boo8509281623719168167x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_528
tff(fact_10651_ATP_Olambda__529,axiom,
    ! [Uu: set_complex,Uua: fun_na5941084329111710283x_bool,Uub: nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aa_fun2551091230351774588x_bool(aTP_Lamm_lw(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_nat5545486172511849908x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_529
tff(fact_10652_ATP_Olambda__530,axiom,
    ! [Uu: set_complex,Uua: fun_in5750378413879446255x_bool,Uub: int,Uuc: complex] :
      ( pp(aa_complex_bool(aa_int5354780257279585880x_bool(aa_fun6456964791495427196x_bool(aTP_Lamm_mb(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_int5354780257279585880x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_530
tff(fact_10653_ATP_Olambda__531,axiom,
    ! [Uu: set_real,Uua: fun_re1255401033344146673l_bool,Uub: real,Uuc: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aa_fun1425803159125577980l_bool(aTP_Lamm_ra(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_531
tff(fact_10654_ATP_Olambda__532,axiom,
    ! [Uu: set_real,Uua: fun_bo2025495424286513072l_bool,Uub: bool,Uuc: real] :
      ( pp(aa_real_bool(aa_boo2589871674041770537l_bool(aa_fun1574454591505385630l_bool(aTP_Lamm_qu(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_real_bool(aa_boo2589871674041770537l_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_532
tff(fact_10655_ATP_Olambda__533,axiom,
    ! [Uu: set_bool,Uua: fun_re3316061561565328172l_bool,Uub: real,Uuc: bool] :
      ( pp(aa_bool_bool(aa_rea6668032140610767853l_bool(aa_fun1775789228971148182l_bool(aTP_Lamm_qx(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_bool_bool(member_bool(Uuc),Uu))
        & pp(aa_bool_bool(aa_rea6668032140610767853l_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_533
tff(fact_10656_ATP_Olambda__534,axiom,
    ! [Uu: set_bool,Uua: fun_bo1841576699660343469l_bool,Uub: bool,Uuc: bool] :
      ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(aa_fun1447330757145981436l_bool(aTP_Lamm_qr(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_bool_bool(member_bool(Uuc),Uu))
        & pp(aa_bool_bool(aa_boo8970175072276269406l_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_534
tff(fact_10657_ATP_Olambda__535,axiom,
    ! [Uu: set_nat,Uua: fun_Ex7682446853002821117t_bool,Uub: extended_enat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_Ext6309394871002001526t_bool(aa_fun7698690945434549304t_bool(aTP_Lamm_lj(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_Ext6309394871002001526t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_535
tff(fact_10658_ATP_Olambda__536,axiom,
    ! [Uu: set_nat,Uua: fun_co2841405903858895691t_bool,Uub: complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_le(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_com5335841084339320244t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_536
tff(fact_10659_ATP_Olambda__537,axiom,
    ! [Uu: set_nat,Uua: fun_re9121486324101951821t_bool,Uub: real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_kv(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_real_fun_nat_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_537
tff(fact_10660_ATP_Olambda__538,axiom,
    ! [Uu: set_nat,Uua: fun_bo3288154511880838028t_bool,Uub: bool,Uuc: nat] :
      ( pp(aa_nat_bool(aa_bool_fun_nat_bool(aa_fun7537470554201192406t_bool(aTP_Lamm_kr(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_bool_fun_nat_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_538
tff(fact_10661_ATP_Olambda__539,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_bool,Uub: int,Uuc: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(aa_fun3602880909931171836t_bool(aTP_Lamm_kz(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_539
tff(fact_10662_ATP_Olambda__540,axiom,
    ! [Uu: set_int,Uua: fun_re8661203372455305201t_bool,Uub: real,Uuc: int] :
      ( pp(aa_int_bool(aa_real_fun_int_bool(aa_fun6227302519962330364t_bool(aTP_Lamm_re(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_real_fun_int_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_540
tff(fact_10663_ATP_Olambda__541,axiom,
    ! [Uu: set_int,Uua: fun_bo2827871560234191408t_bool,Uub: bool,Uuc: int] :
      ( pp(aa_int_bool(aa_bool_fun_int_bool(aa_fun4998116011708744478t_bool(aTP_Lamm_rc(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_bool_fun_int_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_541
tff(fact_10664_ATP_Olambda__542,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex650070010719229343x_bool,Uub: complex,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_com1925010942101449184t_bool(aa_fun1028883470959682428t_bool(aTP_Lamm_mj(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu))
        & pp(aa_complex_bool(aa_Ext6408740497486688536x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_542
tff(fact_10665_ATP_Olambda__543,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex7682446853002821117t_bool,Uub: nat,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_nat6328455298437991426t_bool(aa_fun955218191390214396t_bool(aTP_Lamm_lm(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu))
        & pp(aa_nat_bool(aa_Ext6309394871002001526t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_543
tff(fact_10666_ATP_Olambda__544,axiom,
    ! [Uu: set_complex,Uua: fun_co2841405903858895691t_bool,Uub: nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_lh(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_nat_bool(aa_com5335841084339320244t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_544
tff(fact_10667_ATP_Olambda__545,axiom,
    ! [Uu: set_real,Uua: fun_re7357811994743134493t_bool,Uub: extended_enat,Uuc: real] :
      ( pp(aa_real_bool(aa_Ext4143593917056713754l_bool(aa_fun3666023268991729404l_bool(aTP_Lamm_dk(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_Ext2772979874139974381t_bool(aa_rea3571252896447469150t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_545
tff(fact_10668_ATP_Olambda__546,axiom,
    ! [Uu: set_real,Uua: fun_re3010880487650781423x_bool,Uub: complex,Uuc: real] :
      ( pp(aa_real_bool(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_di(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_546
tff(fact_10669_ATP_Olambda__547,axiom,
    ! [Uu: set_real,Uua: fun_re1255401033344146673l_bool,Uub: real,Uuc: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aa_fun1425803159125577980l_bool(aTP_Lamm_qz(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_547
tff(fact_10670_ATP_Olambda__548,axiom,
    ! [Uu: set_real,Uua: fun_re3316061561565328172l_bool,Uub: bool,Uuc: real] :
      ( pp(aa_real_bool(aa_boo2589871674041770537l_bool(aa_fun485223091692333082l_bool(aTP_Lamm_qw(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_bool_bool(aa_rea6668032140610767853l_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_548
tff(fact_10671_ATP_Olambda__549,axiom,
    ! [Uu: set_real,Uua: fun_re9121486324101951821t_bool,Uub: nat,Uuc: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_de(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_nat_bool(aa_real_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_549
tff(fact_10672_ATP_Olambda__550,axiom,
    ! [Uu: set_real,Uua: fun_re8661203372455305201t_bool,Uub: int,Uuc: real] :
      ( pp(aa_real_bool(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_dg(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_int_bool(aa_real_fun_int_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_550
tff(fact_10673_ATP_Olambda__551,axiom,
    ! [Uu: set_bool,Uua: fun_bo6623066019567229854t_bool,Uub: extended_enat,Uuc: bool] :
      ( pp(aa_bool_bool(aa_Ext5859091071952444205l_bool(aa_fun4004702039914900634l_bool(aTP_Lamm_dj(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_bool_bool(member_bool(Uuc),Uu))
        & pp(aa_Ext2772979874139974381t_bool(aa_boo1364755444788961487t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_551
tff(fact_10674_ATP_Olambda__552,axiom,
    ! [Uu: set_bool,Uua: fun_bo8977660800532043054x_bool,Uub: complex,Uuc: bool] :
      ( pp(aa_bool_bool(aa_com4332222969190720879l_bool(aa_fun6951469949174074266l_bool(aTP_Lamm_dh(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_bool_bool(member_bool(Uuc),Uu))
        & pp(aa_complex_bool(aa_boo8509281623719168167x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_552
tff(fact_10675_ATP_Olambda__553,axiom,
    ! [Uu: set_bool,Uua: fun_bo2025495424286513072l_bool,Uub: real,Uuc: bool] :
      ( pp(aa_bool_bool(aa_rea6668032140610767853l_bool(aa_fun2865020728784200730l_bool(aTP_Lamm_qt(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_bool_bool(member_bool(Uuc),Uu))
        & pp(aa_real_bool(aa_boo2589871674041770537l_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_553
tff(fact_10676_ATP_Olambda__554,axiom,
    ! [Uu: set_bool,Uua: fun_bo1841576699660343469l_bool,Uub: bool,Uuc: bool] :
      ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(aa_fun1447330757145981436l_bool(aTP_Lamm_qq(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_bool_bool(member_bool(Uuc),Uu))
        & pp(aa_bool_bool(aa_boo8970175072276269406l_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_554
tff(fact_10677_ATP_Olambda__555,axiom,
    ! [Uu: set_bool,Uua: fun_bo3288154511880838028t_bool,Uub: nat,Uuc: bool] :
      ( pp(aa_bool_bool(aa_nat_fun_bool_bool(aa_fun4163026448321506586l_bool(aTP_Lamm_dd(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_bool_bool(member_bool(Uuc),Uu))
        & pp(aa_nat_bool(aa_bool_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_555
tff(fact_10678_ATP_Olambda__556,axiom,
    ! [Uu: set_bool,Uua: fun_bo2827871560234191408t_bool,Uub: int,Uuc: bool] :
      ( pp(aa_bool_bool(aa_int_fun_bool_bool(aa_fun6035399485891414554l_bool(aTP_Lamm_df(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_bool_bool(member_bool(Uuc),Uu))
        & pp(aa_int_bool(aa_bool_fun_int_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_556
tff(fact_10679_ATP_Olambda__557,axiom,
    ! [Uu: set_nat,Uua: fun_na5941084329111710283x_bool,Uub: complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_lz(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_complex_bool(aa_nat5545486172511849908x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_557
tff(fact_10680_ATP_Olambda__558,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_bool,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_dl(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_558
tff(fact_10681_ATP_Olambda__559,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_bool,Uub: int,Uuc: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_dm(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_int_bool(aa_nat_fun_int_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_559
tff(fact_10682_ATP_Olambda__560,axiom,
    ! [Uu: set_int,Uua: fun_in5750378413879446255x_bool,Uub: complex,Uuc: int] :
      ( pp(aa_int_bool(aa_com2211550117870696024t_bool(aa_fun3087709329828230012t_bool(aTP_Lamm_me(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_complex_bool(aa_int5354780257279585880x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_560
tff(fact_10683_ATP_Olambda__561,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_bool,Uub: nat,Uuc: int] :
      ( pp(aa_int_bool(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_lc(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_561
tff(fact_10684_ATP_Olambda__562,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_sp(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu),Uuc)),aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_562
tff(fact_10685_ATP_Olambda__563,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ss(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_563
tff(fact_10686_ATP_Olambda__564,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_sq(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Uu),Uuc)),aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_564
tff(fact_10687_ATP_Olambda__565,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_sr(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu),Uuc)),aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_565
tff(fact_10688_ATP_Olambda__566,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_wz(Uu),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ) ).

% ATP.lambda_566
tff(fact_10689_ATP_Olambda__567,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_wm(Uu),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_567
tff(fact_10690_ATP_Olambda__568,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_xb(Uu),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ) ).

% ATP.lambda_568
tff(fact_10691_ATP_Olambda__569,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_wo(Uu),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_569
tff(fact_10692_ATP_Olambda__570,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex3741802288643809319omplex,Uub: fun_Ex3741802288643809319omplex,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun7434105802671982559t_bool(aa_fun6445447259835133315t_bool(aTP_Lamm_in(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_Ext8748583412541189208omplex(Uua,Uuc)),aa_Ext8748583412541189208omplex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_570
tff(fact_10693_ATP_Olambda__571,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_il(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_571
tff(fact_10694_ATP_Olambda__572,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_if(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_572
tff(fact_10695_ATP_Olambda__573,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_ir(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_573
tff(fact_10696_ATP_Olambda__574,axiom,
    ! [Uu: set_bool,Uua: fun_bool_complex,Uub: fun_bool_complex,Uuc: bool] :
      ( pp(aa_bool_bool(aa_fun4000492451996535199l_bool(aa_fun492156817397607122l_bool(aTP_Lamm_id(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_bool_bool(member_bool(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_bool_complex(Uua,Uuc)),aa_bool_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_574
tff(fact_10697_ATP_Olambda__575,axiom,
    ! [Uu: set_bool,Uua: fun_bool_real,Uub: fun_bool_real,Uuc: bool] :
      ( pp(aa_bool_bool(aa_fun2512541629346179229l_bool(aa_fun8125513006060278610l_bool(aTP_Lamm_ip(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_bool_bool(member_bool(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_bool_real(Uua,Uuc)),aa_bool_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_575
tff(fact_10698_ATP_Olambda__576,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_ih(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_576
tff(fact_10699_ATP_Olambda__577,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_it(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_577
tff(fact_10700_ATP_Olambda__578,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_ij(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_578
tff(fact_10701_ATP_Olambda__579,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_iv(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_579
tff(fact_10702_ATP_Olambda__580,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: fun_Ex1451620480498224165t_real,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun338017137003533149t_bool(aa_fun6293361260138588547t_bool(aTP_Lamm_jh(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Ext2541572767299617110t_real(Uua,Uuc)),aa_Ext2541572767299617110t_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_580
tff(fact_10703_ATP_Olambda__581,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_jf(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_581
tff(fact_10704_ATP_Olambda__582,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_iz(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(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_582
tff(fact_10705_ATP_Olambda__583,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_jl(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_583
tff(fact_10706_ATP_Olambda__584,axiom,
    ! [Uu: set_bool,Uua: fun_bool_real,Uub: fun_bool_real,Uuc: bool] :
      ( pp(aa_bool_bool(aa_fun2512541629346179229l_bool(aa_fun8125513006060278610l_bool(aTP_Lamm_ix(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_bool_bool(member_bool(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_bool_real(Uua,Uuc)),aa_bool_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_584
tff(fact_10707_ATP_Olambda__585,axiom,
    ! [Uu: set_bool,Uua: fun_bool_rat,Uub: fun_bool_rat,Uuc: bool] :
      ( pp(aa_bool_bool(aa_fun4510340986217973961l_bool(aa_fun5608642267463484754l_bool(aTP_Lamm_jj(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_bool_bool(member_bool(Uuc),Uu))
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_bool_rat(Uua,Uuc)),aa_bool_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_585
tff(fact_10708_ATP_Olambda__586,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_jb(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_586
tff(fact_10709_ATP_Olambda__587,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_jn(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_587
tff(fact_10710_ATP_Olambda__588,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_jd(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_588
tff(fact_10711_ATP_Olambda__589,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      ( pp(aa_int_bool(aa_fun258058769140594373t_bool(aa_fun465274253410646749t_bool(aTP_Lamm_jp(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_589
tff(fact_10712_ATP_Olambda__590,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_tq(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_590
tff(fact_10713_ATP_Olambda__591,axiom,
    ! [Uu: set_complex,Uua: fun_Ex6753287174829011282omplex,Uub: fun_Ex650070010719229343x_bool,Uuc: extended_enat] : aa_Ext8748583412541189208omplex(aa_fun5203025040472162314omplex(aa_fun6421059515561147915omplex(aTP_Lamm_mh(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_Ext6914378186415866627omplex(Uua,Uuc),collect_complex(aa_Ext6408740497486688536x_bool(aa_fun5941807508050755708x_bool(aTP_Lamm_mg(Uu),Uub),Uuc))) ).

% ATP.lambda_591
tff(fact_10714_ATP_Olambda__592,axiom,
    ! [Uu: set_complex,Uua: fun_in1307110748760623106omplex,Uub: fun_in5750378413879446255x_bool,Uuc: int] : aa_int_complex(aa_fun751309451324587018omplex(aa_fun3438240386181387033omplex(aTP_Lamm_mc(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_int3545933452204062147omplex(Uua,Uuc),collect_complex(aa_int5354780257279585880x_bool(aa_fun6456964791495427196x_bool(aTP_Lamm_mb(Uu),Uub),Uuc))) ).

% ATP.lambda_592
tff(fact_10715_ATP_Olambda__593,axiom,
    ! [Uu: set_complex,Uua: fun_na1591161823130732966omplex,Uub: fun_na5941084329111710283x_bool,Uuc: nat] : aa_nat_complex(aa_fun2700696402887535882omplex(aa_fun8019143150332516725omplex(aTP_Lamm_lx(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(Uua,Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun2551091230351774588x_bool(aTP_Lamm_lw(Uu),Uub),Uuc))) ).

% ATP.lambda_593
tff(fact_10716_ATP_Olambda__594,axiom,
    ! [Uu: set_complex,Uua: fun_re7208826350674946562omplex,Uub: fun_re3010880487650781423x_bool,Uuc: real] : aa_real_complex(aa_fun4873726854752153866omplex(aa_fun883667864959905433omplex(aTP_Lamm_lt(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_rea2618301233319580611omplex(Uua,Uuc),collect_complex(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_ls(Uu),Uub),Uuc))) ).

% ATP.lambda_594
tff(fact_10717_ATP_Olambda__595,axiom,
    ! [Uu: set_complex,Uua: fun_bo6990880009230973955omplex,Uub: fun_bo8977660800532043054x_bool,Uuc: bool] : aa_bool_complex(aa_fun7507808871734033644omplex(aa_fun3415237024035613564omplex(aTP_Lamm_lp(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_boo4396326699157026228omplex(Uua,Uuc),collect_complex(aa_boo8509281623719168167x_bool(aa_fun8473158556963620890x_bool(aTP_Lamm_lo(Uu),Uub),Uuc))) ).

% ATP.lambda_595
tff(fact_10718_ATP_Olambda__596,axiom,
    ! [Uu: set_nat,Uua: fun_Ex7123704652682714774at_nat,Uub: fun_Ex7682446853002821117t_bool,Uuc: extended_enat] : aa_Extended_enat_nat(aa_fun6947057244751970126at_nat(aa_fun635179117725900299at_nat(aTP_Lamm_lk(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_Ext158989295719858759at_nat(Uua,Uuc)),collect_nat(aa_Ext6309394871002001526t_bool(aa_fun7698690945434549304t_bool(aTP_Lamm_lj(Uu),Uub),Uuc))) ).

% ATP.lambda_596
tff(fact_10719_ATP_Olambda__597,axiom,
    ! [Uu: set_nat,Uua: fun_co636942507509095880at_nat,Uub: fun_co2841405903858895691t_bool,Uuc: complex] : aa_complex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_lf(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_com6383720667567721865at_nat(Uua,Uuc)),collect_nat(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_le(Uu),Uub),Uuc))) ).

% ATP.lambda_597
tff(fact_10720_ATP_Olambda__598,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_bool,Uuc: int] : aa_int_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_la(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_bool(aa_fun3602880909931171836t_bool(aTP_Lamm_kz(Uu),Uub),Uuc))) ).

% ATP.lambda_598
tff(fact_10721_ATP_Olambda__599,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_re9121486324101951821t_bool,Uuc: real] : aa_real_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_kw(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_real_fun_nat_nat(Uua,Uuc)),collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_kv(Uu),Uub),Uuc))) ).

% ATP.lambda_599
tff(fact_10722_ATP_Olambda__600,axiom,
    ! [Uu: set_nat,Uua: fun_bool_fun_nat_nat,Uub: fun_bo3288154511880838028t_bool,Uuc: bool] : aa_bool_nat(aa_fun7928520491197679664ol_nat(aa_fun5318524179215677308ol_nat(aTP_Lamm_ks(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_bool_fun_nat_nat(Uua,Uuc)),collect_nat(aa_bool_fun_nat_bool(aa_fun7537470554201192406t_bool(aTP_Lamm_kr(Uu),Uub),Uuc))) ).

% ATP.lambda_600
tff(fact_10723_ATP_Olambda__601,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_op(Uu),Uua),Uub),Uuc) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_601
tff(fact_10724_ATP_Olambda__602,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_oo(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_602
tff(fact_10725_ATP_Olambda__603,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_vc(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_vb(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_603
tff(fact_10726_ATP_Olambda__604,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_vx(Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uuc),Uud))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uud),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu)))
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))
           => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(Uua,I4)),X_1))
            <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Uub),I4)) ) )
        & ( ( Uuc = Uud )
         => ! [X3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Uua)))
             => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) ) )
        & ( ( Uuc != Uud )
         => ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),Uud))
            & ! [X3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu)))
               => ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),X3))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),X3))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Uud)) ) ) ) ) ) ) ) ).

% ATP.lambda_604
tff(fact_10727_ATP_Olambda__605,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hn,Uu),Uua) = Uu ).

% ATP.lambda_605
tff(fact_10728_ATP_Olambda__606,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_rt,Uu) = Uu ).

% ATP.lambda_606
tff(fact_10729_ATP_Olambda__607,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_qb,Uu) = Uu ).

% ATP.lambda_607
tff(fact_10730_ATP_Olambda__608,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ru,Uu) = Uu ).

% ATP.lambda_608
tff(fact_10731_ATP_Olambda__609,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_bz,Uu) = zero_zero_complex ).

% ATP.lambda_609
tff(fact_10732_ATP_Olambda__610,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ca,Uu) = zero_zero_real ).

% ATP.lambda_610
tff(fact_10733_ATP_Olambda__611,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_cb,Uu) = zero_zero_rat ).

% ATP.lambda_611
tff(fact_10734_ATP_Olambda__612,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_gy,Uu) = zero_zero_real ).

% ATP.lambda_612
tff(fact_10735_ATP_Olambda__613,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_cc,Uu) = zero_zero_nat ).

% ATP.lambda_613
tff(fact_10736_ATP_Olambda__614,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_gz,Uu) = zero_zero_int ).

% ATP.lambda_614
tff(fact_10737_ATP_Olambda__615,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_cd,Uu) = zero_zero_int ).

% ATP.lambda_615
tff(fact_10738_ATP_Olambda__616,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_er,Uu))
    <=> $false ) ).

% ATP.lambda_616
tff(fact_10739_ATP_Olambda__617,axiom,
    ! [Uu: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_ep,Uu))
    <=> $false ) ).

% ATP.lambda_617
tff(fact_10740_ATP_Olambda__618,axiom,
    ! [Uu: bool] :
      ( pp(aa_bool_bool(aTP_Lamm_es,Uu))
    <=> $false ) ).

% ATP.lambda_618
tff(fact_10741_ATP_Olambda__619,axiom,
    ! [Uu: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_eq,Uu))
    <=> $false ) ).

% ATP.lambda_619
tff(fact_10742_ATP_Olambda__620,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_et,Uu))
    <=> $false ) ).

% ATP.lambda_620
tff(fact_10743_ATP_Olambda__621,axiom,
    ! [Uu: int] :
      ( pp(aa_int_bool(aTP_Lamm_eu,Uu))
    <=> $false ) ).

% ATP.lambda_621
tff(fact_10744_ATP_Olambda__622,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_xr,Uu))
    <=> $true ) ).

% ATP.lambda_622

% Helper facts (70)
tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( pp(P)
      | pp(aa_bool_bool(fNot,P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ pp(aa_bool_bool(fNot,P))
      | ~ pp(P) ) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,P),Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,P),Q))
      | pp(P) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | ~ pp(Q)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,P),Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,P),Q))
      | pp(P)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,P),Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,P),Q)) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q))
      | ~ pp(P)
      | pp(Q) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( pp(P)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q)) ) ).

tff(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(if_int(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(if_int(fTrue),X),Y) = X ).

tff(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fTrue),X),Y) = X ).

tff(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(if_num(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(if_num(fTrue),X),Y) = X ).

tff(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(if_rat(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(if_rat(fTrue),X),Y) = X ).

tff(help_fEx_1_1_fEx_001t__Nat__Onat_U,axiom,
    ! [P: fun_nat_bool,X: nat] :
      ( ~ pp(aa_nat_bool(P,X))
      | pp(aa_fun_nat_bool_bool(fEx_nat,P)) ) ).

tff(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(if_real(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(if_real(fTrue),X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ( X != Y )
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y))
      | ( X = Y ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_bool] : aa_real_bool(P,fChoice_real(P)) = aa_fun3352386287069783311l_bool(fEx_real,P) ).

tff(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(if_complex(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(if_complex(fTrue,X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( X != Y )
      | pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(if_Extended_enat(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(if_Extended_enat(fTrue,X),Y) = X ).

tff(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] : if_Code_integer(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] : if_Code_integer(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] : if_VEBT_VEBT(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] : if_VEBT_VEBT(fTrue,X,Y) = X ).

tff(help_fequal_2_1_fequal_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( X != Y )
      | pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ pp(aa_Ext2772979874139974381t_bool(aa_Ext291666158496884126t_bool(fequal_Extended_enat,X),Y))
      | ( X = Y ) ) ).

tff(help_fAll_1_1_fAll_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( ~ pp(fAll_VEBT_VEBT(P))
      | pp(aa_VEBT_VEBT_bool(P,X)) ) ).

tff(help_fequal_2_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
      | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X != Y )
      | pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(fequal_set_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(fequal_set_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] : if_option_nat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] : if_option_nat(fTrue,X,Y) = X ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( X != Y )
      | pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(fequal_list_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ~ pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(fequal_list_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( X != Y )
      | pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(fTrue),X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( X != Y )
      | pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(fequal9188845342289841878at_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ~ pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(fequal9188845342289841878at_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_COMBB_1_1_COMBB_001t__HOL__Obool_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_bool_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_3745307990312268622T_VEBT(P,Q),R) = aa_bool_bool(P,aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool_001t__HOL__Obool_U,axiom,
    ! [P: fun_VE2066785035971159626l_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBS_6222329278419335549l_bool(P,Q),R) = aa_bool_bool(aa_VEB8047906998784800507l_bool(P,R),aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fTrue,X,Y) = X ).

tff(help_COMBB_1_1_COMBB_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_fu7168253649841883962l_bool,Q: fun_VE3763956218935105199t_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_2885587777813823575T_VEBT(P,Q),R) = aa_fun_nat_bool_bool(P,aa_VEB4832260192069861672t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_001t__HOL__Obool_001t__fun_It__HOL__Obool_Mt__HOL__Obool_J_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_bo1841576699660343469l_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEB8047906998784800507l_bool(cOMBB_3438363824644082366T_VEBT(P,Q),R) = aa_boo8970175072276269406l_bool(P,aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_001t__HOL__Obool_U,axiom,
    ! [P: fun_VE7346385682144900493T_bool,Q: set_VEBT_VEBT,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBC_771164880915498576T_bool(P,Q),R) = aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(P,R),Q) ).

% Conjectures (4)
tff(conj_0,hypothesis,
    ~ ( ( vEBT_vebt_mint(ta) = none_nat )
      & ( vEBT_vebt_mint(k) = some_nat(b) ) ) ).

tff(conj_1,hypothesis,
    ~ ( ( vEBT_vebt_mint(ta) = some_nat(a) )
      & ( vEBT_vebt_mint(k) = none_nat ) ) ).

tff(conj_2,hypothesis,
    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,a),b))
    & ( some_nat(a) = vEBT_vebt_mint(ta) )
    & ( some_nat(b) = vEBT_vebt_mint(k) ) ) ).

tff(conj_3,conjecture,
    $false ).

%------------------------------------------------------------------------------